Theorem cldval 14267

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 14176	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3		pwexg 4209	. . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
4		rabexg 4172	. . 3 ⊢ (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ∈ V)
5	2, 3, 4	3syl 17	. 2 ⊢ (𝐽 ∈ Top → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ∈ V)
6		unieq 3844	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
7	6, 1	eqtr4di 2244	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
8	7	pweqd 3606	. . . 4 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
9	7	difeq1d 3276	. . . . 5 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 ∖ 𝑥) = (𝑋 ∖ 𝑥))
10		eleq12 2258	. . . . 5 ⊢ (((∪ 𝑗 ∖ 𝑥) = (𝑋 ∖ 𝑥) ∧ 𝑗 = 𝐽) → ((∪ 𝑗 ∖ 𝑥) ∈ 𝑗 ↔ (𝑋 ∖ 𝑥) ∈ 𝐽))
11	9, 10	mpancom 422	. . . 4 ⊢ (𝑗 = 𝐽 → ((∪ 𝑗 ∖ 𝑥) ∈ 𝑗 ↔ (𝑋 ∖ 𝑥) ∈ 𝐽))
12	8, 11	rabeqbidv 2755	. . 3 ⊢ (𝑗 = 𝐽 → {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})
13		df-cld 14263	. . 3 ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
14	12, 13	fvmptg 5633	. 2 ⊢ ((𝐽 ∈ Top ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ∈ V) → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})
15	5, 14	mpdan 421	1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {crab 2476  Vcvv 2760   ∖ cdif 3150  𝒫 cpw 3601  ∪ cuni 3835  ‘cfv 5254  Topctop 14165  Clsdccld 14260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-top 14166  df-cld 14263

This theorem is referenced by:  iscld  14271