Theorem opncldf3 23115

Description: The values of the converse/inverse 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
opncldf3	⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = (𝑋 ∖ 𝐵))

Distinct variable groups:   𝑢,𝐽   𝑢,𝑋

Allowed substitution hints:   𝐵(𝑢)   𝐹(𝑢)

Proof of Theorem opncldf3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cldrcl 23055	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		opncldf.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3		opncldf.2	. . . . . 6 ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))
4	2, 3	opncldf1 23113	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))))
5	4	simprd 498	. . . 4 ⊢ (𝐽 ∈ Top → ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)))
6	1, 5	syl 17	. . 3 ⊢ (𝐵 ∈ (Clsd‘𝐽) → ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)))
7	6	fveq1d 6854	. 2 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵))
8	2	cldopn 23060	. . 3 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐵) ∈ 𝐽)
9		difeq2 4065	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝐵))
10		eqid 2752	. . . 4 ⊢ (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)) = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))
11	9, 10	fvmptg 6958	. . 3 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝑋 ∖ 𝐵) ∈ 𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵) = (𝑋 ∖ 𝐵))
12	8, 11	mpdan 695	. 2 ⊢ (𝐵 ∈ (Clsd‘𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵) = (𝑋 ∖ 𝐵))
13	7, 12	eqtrd 2787	1 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = (𝑋 ∖ 𝐵))

Syntax hints:   → wi 4   = wceq 1550   ∈ wcel 2132   ∖ cdif 3892  ∪ cuni 4855   ↦ cmpt 5171  ^◡ccnv 5635  –1-1-onto→wf1o 6505  ‘cfv 6506  Topctop 22922  Clsdccld 23045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-top 22923  df-cld 23048

This theorem is referenced by: (None)