Theorem cldssbrsiga 34223

Description: A Borel Algebra contains all closed sets of its base topology. (Contributed by Thierry Arnoux, 27-Mar-2017.)

Assertion

Ref	Expression
cldssbrsiga	⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))

Proof of Theorem cldssbrsiga

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldss 22972	. . . . . 6 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ ∪ 𝐽)
3	2	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ ∪ 𝐽)
4		dfss4 4249	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
5	3, 4	sylib 218	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
6	1	topopn 22849	. . . . . 6 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
7	1	difopn 22977	. . . . . 6 ⊢ ((∪ 𝐽 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽)
8	6, 7	sylan 580	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽)
9		id 22	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
10	9	sgsiga 34178	. . . . . . 7 ⊢ (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
11	10	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
12		elex 3485	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V)
13		sigagensiga 34177	. . . . . . . 8 ⊢ (𝐽 ∈ V → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽))
14		baselsiga 34151	. . . . . . . 8 ⊢ ((sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽) → ∪ 𝐽 ∈ (sigaGen‘𝐽))
15	12, 13, 14	3syl 18	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ (sigaGen‘𝐽))
16	15	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → ∪ 𝐽 ∈ (sigaGen‘𝐽))
17		elsigagen 34183	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (sigaGen‘𝐽))
18		difelsiga 34169	. . . . . 6 ⊢ (((sigaGen‘𝐽) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐽 ∈ (sigaGen‘𝐽) ∧ (∪ 𝐽 ∖ 𝑥) ∈ (sigaGen‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽))
19	11, 16, 17, 18	syl3anc 1373	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽))
20	8, 19	syldan 591	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽))
21	5, 20	eqeltrrd 2836	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (sigaGen‘𝐽))
22	21	ex 412	. 2 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ (sigaGen‘𝐽)))
23	22	ssrdv 3969	1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∖ cdif 3928   ⊆ wss 3931  ∪ cuni 4888  ran crn 5660  ‘cfv 6536  Topctop 22836  Clsdccld 22959  sigAlgebracsiga 34144  sigaGencsigagen 34174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-ac2 10482

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9529  df-dju 9920  df-card 9958  df-acn 9961  df-ac 10135  df-top 22837  df-cld 22962  df-siga 34145  df-sigagen 34175

This theorem is referenced by:  sxbrsigalem4  34324  sibfinima  34376  sibfof  34377  orvccel  34500