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Theorem cldssbrsiga 29577
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.
StepHypRef Expression
1 eqid 2610 . . . . . . 7 𝐽 = 𝐽
21cldss 20643 . . . . . 6 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 𝐽)
32adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 𝐽)
4 dfss4 3820 . . . . 5 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
53, 4sylib 207 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
61topopn 20536 . . . . . 6 (𝐽 ∈ Top → 𝐽𝐽)
71difopn 20648 . . . . . 6 (( 𝐽𝐽𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ 𝐽)
86, 7sylan 487 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ 𝐽)
9 id 22 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 ∈ Top)
109sgsiga 29532 . . . . . . 7 (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ ran sigAlgebra)
1110adantr 480 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → (sigaGen‘𝐽) ∈ ran sigAlgebra)
12 elex 3185 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 ∈ V)
13 sigagensiga 29531 . . . . . . . 8 (𝐽 ∈ V → (sigaGen‘𝐽) ∈ (sigAlgebra‘ 𝐽))
14 baselsiga 29505 . . . . . . . 8 ((sigaGen‘𝐽) ∈ (sigAlgebra‘ 𝐽) → 𝐽 ∈ (sigaGen‘𝐽))
1512, 13, 143syl 18 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ (sigaGen‘𝐽))
1615adantr 480 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → 𝐽 ∈ (sigaGen‘𝐽))
17 elsigagen 29537 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → ( 𝐽𝑥) ∈ (sigaGen‘𝐽))
18 difelsiga 29523 . . . . . 6 (((sigaGen‘𝐽) ∈ ran sigAlgebra ∧ 𝐽 ∈ (sigaGen‘𝐽) ∧ ( 𝐽𝑥) ∈ (sigaGen‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
1911, 16, 17, 18syl3anc 1318 . . . . 5 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
208, 19syldan 486 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
215, 20eqeltrrd 2689 . . 3 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (sigaGen‘𝐽))
2221ex 449 . 2 (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ (sigaGen‘𝐽)))
2322ssrdv 3574 1 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cdif 3537  wss 3540   cuni 4372  ran crn 5039  cfv 5804  Topctop 20517  Clsdccld 20630  sigAlgebracsiga 29497  sigaGencsigagen 29528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  df-top 20521  df-cld 20633  df-siga 29498  df-sigagen 29529
This theorem is referenced by:  sxbrsigalem4  29676  sibfinima  29728  sibfof  29729  orvccel  29851
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