Theorem clsfval 14740

Description: The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
cldval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clsfval	⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋

Allowed substitution hint:   𝑋(𝑦)

Proof of Theorem clsfval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cldval.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	topopn 14647	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3		pwexg 4243	. . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
4		mptexg 5837	. . 3 ⊢ (𝒫 𝑋 ∈ V → (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}) ∈ V)
5	2, 3, 4	3syl 17	. 2 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}) ∈ V)
6		unieq 3876	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
7	6, 1	eqtr4di 2260	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
8	7	pweqd 3634	. . . 4 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
9		fveq2 5603	. . . . . 6 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
10		rabeq 2771	. . . . . 6 ⊢ ((Clsd‘𝑗) = (Clsd‘𝐽) → {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦})
11	9, 10	syl 14	. . . . 5 ⊢ (𝑗 = 𝐽 → {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦})
12	11	inteqd 3907	. . . 4 ⊢ (𝑗 = 𝐽 → ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦} = ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦})
13	8, 12	mpteq12dv 4145	. . 3 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}))
14		df-cls 14736	. . 3 ⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}))
15	13, 14	fvmptg 5683	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}) ∈ V) → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}))
16	5, 15	mpdan 421	1 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}))

Syntax hints:   → wi 4   = wceq 1375   ∈ wcel 2180  {crab 2492  Vcvv 2779   ⊆ wss 3177  𝒫 cpw 3629  ∪ cuni 3867  ∩ cint 3902   ↦ cmpt 4124  ‘cfv 5294  Topctop 14636  Clsdccld 14731  clsccl 14733

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-top 14637  df-cls 14736

This theorem is referenced by:  clsval  14750