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Theorem cldssbrsiga 30224
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 2620 . . . . . . 7 𝐽 = 𝐽
21cldss 20814 . . . . . 6 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 𝐽)
32adantl 482 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 𝐽)
4 dfss4 3850 . . . . 5 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
53, 4sylib 208 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
61topopn 20692 . . . . . 6 (𝐽 ∈ Top → 𝐽𝐽)
71difopn 20819 . . . . . 6 (( 𝐽𝐽𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ 𝐽)
86, 7sylan 488 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ 𝐽)
9 id 22 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 ∈ Top)
109sgsiga 30179 . . . . . . 7 (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ ran sigAlgebra)
1110adantr 481 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → (sigaGen‘𝐽) ∈ ran sigAlgebra)
12 elex 3207 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 ∈ V)
13 sigagensiga 30178 . . . . . . . 8 (𝐽 ∈ V → (sigaGen‘𝐽) ∈ (sigAlgebra‘ 𝐽))
14 baselsiga 30152 . . . . . . . 8 ((sigaGen‘𝐽) ∈ (sigAlgebra‘ 𝐽) → 𝐽 ∈ (sigaGen‘𝐽))
1512, 13, 143syl 18 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ (sigaGen‘𝐽))
1615adantr 481 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → 𝐽 ∈ (sigaGen‘𝐽))
17 elsigagen 30184 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → ( 𝐽𝑥) ∈ (sigaGen‘𝐽))
18 difelsiga 30170 . . . . . 6 (((sigaGen‘𝐽) ∈ ran sigAlgebra ∧ 𝐽 ∈ (sigaGen‘𝐽) ∧ ( 𝐽𝑥) ∈ (sigaGen‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
1911, 16, 17, 18syl3anc 1324 . . . . 5 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
208, 19syldan 487 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
215, 20eqeltrrd 2700 . . 3 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (sigaGen‘𝐽))
2221ex 450 . 2 (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ (sigaGen‘𝐽)))
2322ssrdv 3601 1 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cdif 3564  wss 3567   cuni 4427  ran crn 5105  cfv 5876  Topctop 20679  Clsdccld 20801  sigAlgebracsiga 30144  sigaGencsigagen 30175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-card 8750  df-acn 8753  df-ac 8924  df-cda 8975  df-top 20680  df-cld 20804  df-siga 30145  df-sigagen 30176
This theorem is referenced by:  sxbrsigalem4  30323  sibfinima  30375  sibfof  30376  orvccel  30498
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