Theorem opncldf2 21297

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

Hypotheses

Ref	Expression
opncldf.1	⊢ 𝑋 = ∪ 𝐽
opncldf.2	⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))

Assertion

Ref	Expression
opncldf2	⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐹‘𝐴) = (𝑋 ∖ 𝐴))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐽   𝑢,𝑋

Allowed substitution hint:   𝐹(𝑢)

Proof of Theorem opncldf2

Step	Hyp	Ref	Expression
1		simpr 479	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ∈ 𝐽)
2		opncldf.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	2	opncld 21245	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
4		difeq2 3944	. . 3 ⊢ (𝑢 = 𝐴 → (𝑋 ∖ 𝑢) = (𝑋 ∖ 𝐴))
5		opncldf.2	. . 3 ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))
6	4, 5	fvmptg 6540	. 2 ⊢ ((𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽)) → (𝐹‘𝐴) = (𝑋 ∖ 𝐴))
7	1, 3, 6	syl2anc 579	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐹‘𝐴) = (𝑋 ∖ 𝐴))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106   ∖ cdif 3788  ∪ cuni 4671   ↦ cmpt 4965  ‘cfv 6135  Topctop 21105  Clsdccld 21228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-top 21106  df-cld 21231

This theorem is referenced by: (None)