Theorem opncldf3 23094

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 23034	. . . 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 23092	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))))
5	4	simprd 495	. . . 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 6908	. 2 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵))
8	2	cldopn 23039	. . 3 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐵) ∈ 𝐽)
9		difeq2 4120	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝐵))
10		eqid 2737	. . . 4 ⊢ (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)) = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))
11	9, 10	fvmptg 7014	. . 3 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝑋 ∖ 𝐵) ∈ 𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵) = (𝑋 ∖ 𝐵))
12	8, 11	mpdan 687	. 2 ⊢ (𝐵 ∈ (Clsd‘𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵) = (𝑋 ∖ 𝐵))
13	7, 12	eqtrd 2777	1 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = (𝑋 ∖ 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ∖ cdif 3948  ∪ cuni 4907   ↦ cmpt 5225  ^◡ccnv 5684  –1-1-onto→wf1o 6560  ‘cfv 6561  Topctop 22899  Clsdccld 23024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027

This theorem is referenced by: (None)