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Theorems · Theorem · general topology

pathComponentIn_congr

∀ {X : Type u_1} [inst : TopologicalSpace X] {x y : X} {F : Set X},
  x ∈ pathComponentIn F y → pathComponentIn F x = pathComponentIn F y
Defined in
Mathlib.Topology.Connected.PathConnected
Cited by
3 results in Mathlib
Foundations
Depth 127 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
TopologicalSpace
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

IsOpen.pathComponentIn · cited by 5IsOpen.pathComponentInLocallyPathConnectedSpace.coinduced · cited by 2LocallyPathConnectedSpace…locallyPathConnectedSpace_iff_pathComponentIn_mem_nhds · cited by 1locallyPathConnectedSpace…Set · cited by 53352SetTopologicalSpace · cited by 24529TopologicalSpaceSet.ext · cited by 2266Set.extpathComponentIn · cited by 19pathComponentInJoinedIn.trans · cited by 4JoinedIn.transJoinedIn.symm · cited by 3JoinedIn.symmpathComponentIn_congrCITED BYCITES

Cites6

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Cited by3

Results whose statement or proof uses this declaration.