Theorem cldssbrsiga 33937

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 2725	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldss 22977	. . . . . 6 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ ∪ 𝐽)
3	2	adantl 480	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ ∪ 𝐽)
4		dfss4 4257	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
5	3, 4	sylib 217	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
6	1	topopn 22852	. . . . . 6 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
7	1	difopn 22982	. . . . . 6 ⊢ ((∪ 𝐽 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽)
8	6, 7	sylan 578	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽)
9		id 22	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
10	9	sgsiga 33892	. . . . . . 7 ⊢ (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
11	10	adantr 479	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
12		elex 3480	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V)
13		sigagensiga 33891	. . . . . . . 8 ⊢ (𝐽 ∈ V → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽))
14		baselsiga 33865	. . . . . . . 8 ⊢ ((sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽) → ∪ 𝐽 ∈ (sigaGen‘𝐽))
15	12, 13, 14	3syl 18	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ (sigaGen‘𝐽))
16	15	adantr 479	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → ∪ 𝐽 ∈ (sigaGen‘𝐽))
17		elsigagen 33897	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (sigaGen‘𝐽))
18		difelsiga 33883	. . . . . 6 ⊢ (((sigaGen‘𝐽) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐽 ∈ (sigaGen‘𝐽) ∧ (∪ 𝐽 ∖ 𝑥) ∈ (sigaGen‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽))
19	11, 16, 17, 18	syl3anc 1368	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽))
20	8, 19	syldan 589	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽))
21	5, 20	eqeltrrd 2826	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (sigaGen‘𝐽))
22	21	ex 411	. 2 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ (sigaGen‘𝐽)))
23	22	ssrdv 3982	1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ∖ cdif 3941   ⊆ wss 3944  ∪ cuni 4909  ran crn 5679  ‘cfv 6549  Topctop 22839  Clsdccld 22964  sigAlgebracsiga 33858  sigaGencsigagen 33888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-ac2 10488

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9535  df-dju 9926  df-card 9964  df-acn 9967  df-ac 10141  df-top 22840  df-cld 22967  df-siga 33859  df-sigagen 33889

This theorem is referenced by:  sxbrsigalem4  34038  sibfinima  34090  sibfof  34091  orvccel  34213