Theorems · Theorem · order theory
ClosureOperator.setOfPred_isClosed_eq_range_closure
∀ {α : Type u_1} [inst : Preorder α] {c : ClosureOperator α}, {x | c.IsClosed x} = Set.range ⇑cThe set of closed elements for c is exactly its range.
- Defined in
- Mathlib.Order.Closure
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 16 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- Preorderstatement and proof · cited by 7,952
- Set.ofPredstatement and proof · cited by 6,101
- Set.rangestatement and proof · cited by 4,705
- Set.extproof · cited by 2,266
- ClosureOperatorstatement and proof · cited by 371
- ClosureOperator.IsClosedstatement and proof · cited by 38
- ClosureOperator.isClosed_closureproof · cited by 15
- ClosureOperator.IsClosed.closure_eqproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- ClosureOperator.setOf_isClosed_eq_range_closureproof · cited by 0
- LowerAdjoint.closed_eq_range_closeproof · cited by 0