Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cldssbrsiga Structured version   Visualization version   GIF version

Theorem cldssbrsiga 30580
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 2760 . . . . . . 7 𝐽 = 𝐽
21cldss 21055 . . . . . 6 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 𝐽)
32adantl 473 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 𝐽)
4 dfss4 4001 . . . . 5 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
53, 4sylib 208 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
61topopn 20933 . . . . . 6 (𝐽 ∈ Top → 𝐽𝐽)
71difopn 21060 . . . . . 6 (( 𝐽𝐽𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ 𝐽)
86, 7sylan 489 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ 𝐽)
9 id 22 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 ∈ Top)
109sgsiga 30535 . . . . . . 7 (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ ran sigAlgebra)
1110adantr 472 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → (sigaGen‘𝐽) ∈ ran sigAlgebra)
12 elex 3352 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 ∈ V)
13 sigagensiga 30534 . . . . . . . 8 (𝐽 ∈ V → (sigaGen‘𝐽) ∈ (sigAlgebra‘ 𝐽))
14 baselsiga 30508 . . . . . . . 8 ((sigaGen‘𝐽) ∈ (sigAlgebra‘ 𝐽) → 𝐽 ∈ (sigaGen‘𝐽))
1512, 13, 143syl 18 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ (sigaGen‘𝐽))
1615adantr 472 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → 𝐽 ∈ (sigaGen‘𝐽))
17 elsigagen 30540 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → ( 𝐽𝑥) ∈ (sigaGen‘𝐽))
18 difelsiga 30526 . . . . . 6 (((sigaGen‘𝐽) ∈ ran sigAlgebra ∧ 𝐽 ∈ (sigaGen‘𝐽) ∧ ( 𝐽𝑥) ∈ (sigaGen‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
1911, 16, 17, 18syl3anc 1477 . . . . 5 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ∈ 𝐽) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
208, 19syldan 488 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ ( 𝐽𝑥)) ∈ (sigaGen‘𝐽))
215, 20eqeltrrd 2840 . . 3 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (sigaGen‘𝐽))
2221ex 449 . 2 (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ (sigaGen‘𝐽)))
2322ssrdv 3750 1 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cdif 3712  wss 3715   cuni 4588  ran crn 5267  cfv 6049  Topctop 20920  Clsdccld 21042  sigAlgebracsiga 30500  sigaGencsigagen 30531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-ac2 9497
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-oi 8582  df-card 8975  df-acn 8978  df-ac 9149  df-cda 9202  df-top 20921  df-cld 21045  df-siga 30501  df-sigagen 30532
This theorem is referenced by:  sxbrsigalem4  30679  sibfinima  30731  sibfof  30732  orvccel  30854
  Copyright terms: Public domain W3C validator