Theorem cldval 22847

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 22728	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3		pwexg 5376	. . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
4		rabexg 5331	. . 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 4919	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
7	6, 1	eqtr4di 2789	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
8	7	pweqd 4619	. . . 4 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
9	7	difeq1d 4121	. . . . 5 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 ∖ 𝑥) = (𝑋 ∖ 𝑥))
10		eleq12 2822	. . . . 5 ⊢ (((∪ 𝑗 ∖ 𝑥) = (𝑋 ∖ 𝑥) ∧ 𝑗 = 𝐽) → ((∪ 𝑗 ∖ 𝑥) ∈ 𝑗 ↔ (𝑋 ∖ 𝑥) ∈ 𝐽))
11	9, 10	mpancom 685	. . . 4 ⊢ (𝑗 = 𝐽 → ((∪ 𝑗 ∖ 𝑥) ∈ 𝑗 ↔ (𝑋 ∖ 𝑥) ∈ 𝐽))
12	8, 11	rabeqbidv 3448	. . 3 ⊢ (𝑗 = 𝐽 → {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})
13		df-cld 22843	. . 3 ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
14	12, 13	fvmptg 6996	. 2 ⊢ ((𝐽 ∈ Top ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ∈ V) → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})
15	5, 14	mpdan 684	1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540   ∈ wcel 2105  {crab 3431  Vcvv 3473   ∖ cdif 3945  𝒫 cpw 4602  ∪ cuni 4908  ‘cfv 6543  Topctop 22715  Clsdccld 22840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-top 22716  df-cld 22843

This theorem is referenced by:  iscld  22851  mretopd  22916