Theorem cldval 21029

Description: The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
cldval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cldval	⊢ (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})

Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem cldval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cldval.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	topopn 20913	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3		pwexg 4999	. . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
4		rabexg 4963	. . 3 ⊢ (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ∈ V)
5	2, 3, 4	3syl 18	. 2 ⊢ (𝐽 ∈ Top → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ∈ V)
6		unieq 4596	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
7	6, 1	syl6eqr 2812	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
8	7	pweqd 4307	. . . 4 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
9	7	difeq1d 3870	. . . . 5 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 ∖ 𝑥) = (𝑋 ∖ 𝑥))
10		eleq12 2829	. . . . 5 ⊢ (((∪ 𝑗 ∖ 𝑥) = (𝑋 ∖ 𝑥) ∧ 𝑗 = 𝐽) → ((∪ 𝑗 ∖ 𝑥) ∈ 𝑗 ↔ (𝑋 ∖ 𝑥) ∈ 𝐽))
11	9, 10	mpancom 706	. . . 4 ⊢ (𝑗 = 𝐽 → ((∪ 𝑗 ∖ 𝑥) ∈ 𝑗 ↔ (𝑋 ∖ 𝑥) ∈ 𝐽))
12	8, 11	rabeqbidv 3335	. . 3 ⊢ (𝑗 = 𝐽 → {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})
13		df-cld 21025	. . 3 ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
14	12, 13	fvmptg 6442	. 2 ⊢ ((𝐽 ∈ Top ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ∈ V) → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})
15	5, 14	mpdan 705	1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632   ∈ wcel 2139  {crab 3054  Vcvv 3340   ∖ cdif 3712  𝒫 cpw 4302  ∪ cuni 4588  ‘cfv 6049  Topctop 20900  Clsdccld 21022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-top 20901  df-cld 21025

This theorem is referenced by:  iscld  21033  mretopd  21098