Theorem opncldf3 21688

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 21628	. . . 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 21686	. . . . 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 6667	. 2 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵))
8	2	cldopn 21633	. . 3 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐵) ∈ 𝐽)
9		difeq2 4093	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝐵))
10		eqid 2821	. . . 4 ⊢ (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)) = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))
11	9, 10	fvmptg 6761	. . 3 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝑋 ∖ 𝐵) ∈ 𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵) = (𝑋 ∖ 𝐵))
12	8, 11	mpdan 685	. 2 ⊢ (𝐵 ∈ (Clsd‘𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵) = (𝑋 ∖ 𝐵))
13	7, 12	eqtrd 2856	1 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = (𝑋 ∖ 𝐵))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110   ∖ cdif 3933  ∪ cuni 4832   ↦ cmpt 5139  ^◡ccnv 5549  –1-1-onto→wf1o 6349  ‘cfv 6350  Topctop 21495  Clsdccld 21618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-top 21496  df-cld 21621

This theorem is referenced by: (None)