Theorems · Theorem · general topology
JoinedIn.joined_subtype
∀ {X : Type u_1} [inst : TopologicalSpace X] {x y : X} {F : Set X} (h : JoinedIn F x y), Joined ⟨x, ⋯⟩ ⟨y, ⋯⟩If x and y are joined in the set F, then they are joined in the subtype F.
- Defined in
- Mathlib.Topology.Connected.PathConnected
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.Elemproof · cited by 7,166
- unitIntervalproof · cited by 607
- ContinuousMapClass.map_continuousproof · cited by 119
- Continuous.subtype_mkproof · cited by 51
- JoinedInstatement and proof · cited by 40
- Joinedstatement · cited by 29
- Path.sourceproof · cited by 26
- Path.targetproof · cited by 24
- JoinedIn.somePathproof · cited by 12
Cited by1
Results whose statement or proof uses this declaration.
- joinedIn_iff_joinedproof · cited by 0