Theorem opncldf3 23404

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 23344	. . . 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 23402	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))))
5	4	simprd 501	. . . 4 ⊢ (𝐽 ∈ Top → ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)))
6	1, 5	syl 18	. . 3 ⊢ (𝐵 ∈ (Clsd‘𝐽) → ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)))
7	6	fveq1d 6887	. 2 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵))
8	2	cldopn 23349	. . 3 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐵) ∈ 𝐽)
9		difeq2 4057	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝐵))
10		eqid 2760	. . . 4 ⊢ (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)) = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))
11	9, 10	fvmptg 6991	. . 3 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝑋 ∖ 𝐵) ∈ 𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵) = (𝑋 ∖ 𝐵))
12	8, 11	mpdan 700	. 2 ⊢ (𝐵 ∈ (Clsd‘𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵) = (𝑋 ∖ 𝐵))
13	7, 12	eqtrd 2795	1 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = (𝑋 ∖ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ∖ cdif 4045  ∪ cuni 4866   ↦ cmpt 5185  ^◡ccnv 5649  –1-1-onto→wf1o 6537  ‘cfv 6538  Topctop 23211  Clsdccld 23334

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-top 23212  df-cld 23337

This theorem is used by: (None)