diff --git a/Mathlib/Geometry/Manifold/ContMDiff/Product.lean b/Mathlib/Geometry/Manifold/ContMDiff/Product.lean
index 6eb4c75f0c438..755c972332459 100644
--- a/Mathlib/Geometry/Manifold/ContMDiff/Product.lean
+++ b/Mathlib/Geometry/Manifold/ContMDiff/Product.lean
@@ -402,8 +402,6 @@ variable {ι : Type*} [Fintype ι] {Fi : ι → Type*} [∀ i, NormedAddCommGrou
 theorem contMDiffWithinAt_pi_space :
     ContMDiffWithinAt I 𝓘(𝕜, ∀ i, Fi i) n φ s x ↔
       ∀ i, ContMDiffWithinAt I 𝓘(𝕜, Fi i) n (fun x => φ x i) s x := by
-  -- Porting note: `simp` fails to apply it on the LHS
-  rw [contMDiffWithinAt_iff]
   simp only [contMDiffWithinAt_iff, continuousWithinAt_pi, contDiffWithinAt_pi, forall_and,
     writtenInExtChartAt, extChartAt_model_space_eq_id, (· ∘ ·), PartialEquiv.refl_coe, id]
 #align cont_mdiff_within_at_pi_space contMDiffWithinAt_pi_space
diff --git a/Mathlib/Geometry/Manifold/Diffeomorph.lean b/Mathlib/Geometry/Manifold/Diffeomorph.lean
index b999247cee358..3385b7b204e14 100644
--- a/Mathlib/Geometry/Manifold/Diffeomorph.lean
+++ b/Mathlib/Geometry/Manifold/Diffeomorph.lean
@@ -570,12 +570,11 @@ with model `I.trans_diffeomorph e`. -/
 def toTransDiffeomorph (e : E ≃ₘ[𝕜] F) : M ≃ₘ⟮I, I.transDiffeomorph e⟯ M where
   toEquiv := Equiv.refl M
   contMDiff_toFun x := by
-    refine' contMDiffWithinAt_iff'.2 ⟨continuousWithinAt_id, _⟩
-    refine' e.contDiff.contDiffWithinAt.congr' (fun y hy => _) _
-    · simp only [Equiv.coe_refl, id, (· ∘ ·), I.coe_extChartAt_transDiffeomorph]
-      -- Porting note: `simp only` failed to used next lemma, converted to `rw`
-      rw [(extChartAt I x).right_inv hy.1]
-    exact
+    refine contMDiffWithinAt_iff'.2 ⟨continuousWithinAt_id, ?_⟩
+    refine e.contDiff.contDiffWithinAt.congr' (fun y hy ↦ ?_) ?_
+    · simp only [Equiv.coe_refl, id, (· ∘ ·), I.coe_extChartAt_transDiffeomorph,
+        (extChartAt I x).right_inv hy.1]
+    · exact
       ⟨(extChartAt I x).map_source (mem_extChartAt_source I x), trivial, by simp only [mfld_simps]⟩
   contMDiff_invFun x := by
     refine' contMDiffWithinAt_iff'.2 ⟨continuousWithinAt_id, _⟩
diff --git a/Mathlib/Geometry/Manifold/VectorBundle/Basic.lean b/Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
index 6d51d71323743..4b698898c9634 100644
--- a/Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
+++ b/Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
@@ -594,9 +594,7 @@ is a smooth vector bundle. -/
 instance smoothVectorBundle : SmoothVectorBundle F Z.Fiber IB where
   smoothOn_coordChangeL := by
     rintro - - ⟨i, rfl⟩ ⟨i', rfl⟩
-    -- Porting note: Originally `Z.smoothOn_coordChange IB i i'`
-    refine'
-      (VectorBundleCore.IsSmooth.smoothOn_coordChange (Z := Z) (IB := IB) i i').congr fun b hb => _
+    refine (Z.smoothOn_coordChange IB i i').congr fun b hb ↦ ?_
     ext v
     exact Z.localTriv_coordChange_eq i i' hb v
 #align vector_bundle_core.smooth_vector_bundle VectorBundleCore.smoothVectorBundle
diff --git a/Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean b/Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean
index ba92f076e9ff9..c2d6497405338 100644
--- a/Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean
+++ b/Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean
@@ -221,21 +221,6 @@ theorem SmoothFiberwiseLinear.locality_aux₂ (e : PartialHomeomorph (B × F) (B
   apply hux
 #align smooth_fiberwise_linear.locality_aux₂ SmoothFiberwiseLinear.locality_aux₂
 
-/- Porting note: `simp only [mem_iUnion]` fails in the next definition. This aux lemma is a
-workaround. -/
-private theorem mem_aux {e : PartialHomeomorph (B × F) (B × F)} :
-    (e ∈ ⋃ (φ : B → F ≃L[𝕜] F) (U : Set B) (hU : IsOpen U)
-      (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => φ x : B → F →L[𝕜] F) U)
-      (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x).symm : B → F →L[𝕜] F) U),
-        {e | e.EqOnSource (FiberwiseLinear.partialHomeomorph φ hU hφ.continuousOn
-          h2φ.continuousOn)}) ↔
-      ∃ (φ : B → F ≃L[𝕜] F) (U : Set B) (hU : IsOpen U)
-        (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => φ x : B → F →L[𝕜] F) U)
-        (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x).symm : B → F →L[𝕜] F) U),
-          e.EqOnSource
-            (FiberwiseLinear.partialHomeomorph φ hU hφ.continuousOn h2φ.continuousOn) := by
-  simp only [mem_iUnion, mem_setOf_eq]
-
 variable (F B IB)
 
 /-- For `B` a manifold and `F` a normed space, the groupoid on `B × F` consisting of local
@@ -249,7 +234,7 @@ def smoothFiberwiseLinear : StructureGroupoid (B × F) where
       (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x).symm : B → F →L[𝕜] F) U),
         {e | e.EqOnSource (FiberwiseLinear.partialHomeomorph φ hU hφ.continuousOn h2φ.continuousOn)}
   trans' := by
-    simp only [mem_aux]
+    simp only [mem_iUnion]
     rintro e e' ⟨φ, U, hU, hφ, h2φ, heφ⟩ ⟨φ', U', hU', hφ', h2φ', heφ'⟩
     refine' ⟨fun b => (φ b).trans (φ' b), _, hU.inter hU', _, _,
       Setoid.trans (PartialHomeomorph.EqOnSource.trans' heφ heφ') ⟨_, _⟩⟩
@@ -272,27 +257,38 @@ def smoothFiberwiseLinear : StructureGroupoid (B × F) where
     simp_rw [ContinuousLinearEquiv.symm_symm]
     exact hφ
   id_mem' := by
-    /- porting note: `simp_rw [mem_iUnion]` failed; expanding. Was:
     simp_rw [mem_iUnion]
-    refine' ⟨fun b => ContinuousLinearEquiv.refl 𝕜 F, univ, isOpen_univ, _, _, ⟨_, fun b hb => _⟩⟩
-    -/
-    refine mem_iUnion.2 ⟨fun _ ↦ .refl 𝕜 F, mem_iUnion.2 ⟨univ, mem_iUnion.2 ⟨isOpen_univ, ?_⟩⟩⟩
-    refine mem_iUnion.2 ⟨contMDiffOn_const, mem_iUnion.2 ⟨contMDiffOn_const, ?_, ?_⟩⟩
-    · simp only [FiberwiseLinear.partialHomeomorph, PartialHomeomorph.refl_partialEquiv,
-        PartialEquiv.refl_source, univ_prod_univ]
-    · exact eqOn_refl id _
+    refine ⟨fun _ ↦ ContinuousLinearEquiv.refl 𝕜 F, univ, isOpen_univ, smoothOn_const,
+      smoothOn_const, ⟨?_, fun b _hb ↦ rfl⟩⟩
+    simp only [FiberwiseLinear.partialHomeomorph, PartialHomeomorph.refl_partialEquiv,
+      PartialEquiv.refl_source, univ_prod_univ]
   locality' := by
     -- the hard work has been extracted to `locality_aux₁` and `locality_aux₂`
-    simp only [mem_aux]
+    simp only [mem_iUnion]
     intro e he
     obtain ⟨U, hU, h⟩ := SmoothFiberwiseLinear.locality_aux₁ e he
     exact SmoothFiberwiseLinear.locality_aux₂ e U hU h
   mem_of_eqOnSource' := by
-    simp only [mem_aux]
+    simp only [mem_iUnion]
     rintro e e' ⟨φ, U, hU, hφ, h2φ, heφ⟩ hee'
     exact ⟨φ, U, hU, hφ, h2φ, Setoid.trans hee' heφ⟩
 #align smooth_fiberwise_linear smoothFiberwiseLinear
 
+variable {F B IB} in
+-- TODO: can this be inlined into the next lemma?
+private theorem mem_aux {e : PartialHomeomorph (B × F) (B × F)} :
+    (e ∈ ⋃ (φ : B → F ≃L[𝕜] F) (U : Set B) (hU : IsOpen U)
+      (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => φ x : B → F →L[𝕜] F) U)
+      (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x).symm : B → F →L[𝕜] F) U),
+        {e | e.EqOnSource (FiberwiseLinear.partialHomeomorph φ hU hφ.continuousOn
+          h2φ.continuousOn)}) ↔
+      ∃ (φ : B → F ≃L[𝕜] F) (U : Set B) (hU : IsOpen U)
+        (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => φ x : B → F →L[𝕜] F) U)
+        (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x).symm : B → F →L[𝕜] F) U),
+          e.EqOnSource
+            (FiberwiseLinear.partialHomeomorph φ hU hφ.continuousOn h2φ.continuousOn) := by
+  simp only [mem_iUnion, mem_setOf_eq]
+
 @[simp]
 theorem mem_smoothFiberwiseLinear_iff (e : PartialHomeomorph (B × F) (B × F)) :
     e ∈ smoothFiberwiseLinear B F IB ↔
diff --git a/Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean b/Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean
index 4e6de44ca1fcd..57f58474f1a4f 100644
--- a/Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean
+++ b/Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean
@@ -376,8 +376,8 @@ theorem tangentBundle_model_space_chartAt (p : TangentBundle I H) :
   ext x : 1
   · ext; · rfl
     exact (tangentBundleCore I H).coordChange_self (achart _ x.1) x.1 (mem_achart_source H x.1) x.2
-  · -- Porting note: was ext; · rfl; apply hEq_of_eq
-    refine congr_arg (TotalSpace.mk _) ?_
+  · ext; · rfl
+    apply heq_of_eq
     exact (tangentBundleCore I H).coordChange_self (achart _ x.1) x.1 (mem_achart_source H x.1) x.2
   simp_rw [TangentBundle.chartAt, FiberBundleCore.localTriv,
     FiberBundleCore.localTrivAsPartialEquiv, VectorBundleCore.toFiberBundleCore_baseSet,