Theorem iscld 22086

Description: The predicate "the class 𝑆 is a closed set". (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
iscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
iscld	⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))

Proof of Theorem iscld

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	cldval 22082	. . . 4 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})
3	2	eleq2d 2824	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽}))
4		difeq2 4047	. . . . 5 ⊢ (𝑥 = 𝑆 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝑆))
5	4	eleq1d 2823	. . . 4 ⊢ (𝑥 = 𝑆 → ((𝑋 ∖ 𝑥) ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ 𝐽))
6	5	elrab 3617	. . 3 ⊢ (𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))
7	3, 6	bitrdi 286	. 2 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))
8	1	topopn 21963	. . . 4 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
9		elpw2g 5263	. . . 4 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
10	8, 9	syl 17	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
11	10	anbi1d 629	. 2 ⊢ (𝐽 ∈ Top → ((𝑆 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))
12	7, 11	bitrd 278	1 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067   ∖ cdif 3880   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  Clsdccld 22075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-top 21951  df-cld 22078

This theorem is referenced by:  iscld2  22087  cldss  22088  cldopn  22090  topcld  22094  discld  22148  indiscld  22150  restcld  22231  ordtcld1  22256  ordtcld2  22257  hauscmp  22466  txcld  22662  ptcld  22672  qtopcld  22772  opnsubg  23167  sszcld  23886  ist0cld  31685  stoweidlem57  43488