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Metamath Proof Explorer

Theorem opncldf2

Description: The values of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009) (Proof shortened by Mario Carneiro, 1-Sep-2015)

Ref Expression
Hypotheses opncldf.1 ⊢ X = ⋃ J
opncldf.2 ⊢ F = u ∈ J ⟼ X ∖ u
Assertion opncldf2 ⊢ J ∈ Top ∧ A ∈ J → F ⁡ A = X ∖ A

Proof

Step Hyp Ref Expression
1 opncldf.1 ⊢ X = ⋃ J
2 opncldf.2 ⊢ F = u ∈ J ⟼ X ∖ u
3 difeq2 ⊢ u = A → X ∖ u = X ∖ A
4 simpr ⊢ J ∈ Top ∧ A ∈ J → A ∈ J
5 1 opncld ⊢ J ∈ Top ∧ A ∈ J → X ∖ A ∈ Clsd ⁡ J
6 2 3 4 5 fvmptd3 ⊢ J ∈ Top ∧ A ∈ J → F ⁡ A = X ∖ A