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Theorem issiga 32751
Description: An alternative definition of the sigma-algebra, for a given base set. (Contributed by Thierry Arnoux, 19-Sep-2016.)
Assertion
Ref Expression
issiga (𝑆 ∈ V β†’ (𝑆 ∈ (sigAlgebraβ€˜π‘‚) ↔ (𝑆 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆)))))
Distinct variable groups:   π‘₯,𝑂   π‘₯,𝑆

Proof of Theorem issiga
Dummy variables π‘œ 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6885 . . . 4 (𝑆 ∈ (sigAlgebraβ€˜π‘‚) β†’ 𝑂 ∈ V)
2 elex 3466 . . . 4 (𝑆 ∈ (sigAlgebraβ€˜π‘‚) β†’ 𝑆 ∈ V)
31, 2jca 513 . . 3 (𝑆 ∈ (sigAlgebraβ€˜π‘‚) β†’ (𝑂 ∈ V ∧ 𝑆 ∈ V))
43a1i 11 . 2 (𝑆 ∈ V β†’ (𝑆 ∈ (sigAlgebraβ€˜π‘‚) β†’ (𝑂 ∈ V ∧ 𝑆 ∈ V)))
5 simpr1 1195 . . . . 5 ((𝑆 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆))) β†’ 𝑂 ∈ 𝑆)
6 elex 3466 . . . . 5 (𝑂 ∈ 𝑆 β†’ 𝑂 ∈ V)
75, 6syl 17 . . . 4 ((𝑆 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆))) β†’ 𝑂 ∈ V)
87a1i 11 . . 3 (𝑆 ∈ V β†’ ((𝑆 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆))) β†’ 𝑂 ∈ V))
98anc2ri 558 . 2 (𝑆 ∈ V β†’ ((𝑆 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆))) β†’ (𝑂 ∈ V ∧ 𝑆 ∈ V)))
10 df-siga 32748 . . . 4 sigAlgebra = (π‘œ ∈ V ↦ {𝑠 ∣ (𝑠 βŠ† 𝒫 π‘œ ∧ (π‘œ ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (π‘œ βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑠)))})
11 sigaex 32749 . . . 4 {𝑠 ∣ (𝑠 βŠ† 𝒫 π‘œ ∧ (π‘œ ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (π‘œ βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑠)))} ∈ V
12 pweq 4579 . . . . . . 7 (π‘œ = 𝑂 β†’ 𝒫 π‘œ = 𝒫 𝑂)
1312sseq2d 3981 . . . . . 6 (π‘œ = 𝑂 β†’ (𝑠 βŠ† 𝒫 π‘œ ↔ 𝑠 βŠ† 𝒫 𝑂))
14 sseq1 3974 . . . . . 6 (𝑠 = 𝑆 β†’ (𝑠 βŠ† 𝒫 𝑂 ↔ 𝑆 βŠ† 𝒫 𝑂))
1513, 14sylan9bb 511 . . . . 5 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ (𝑠 βŠ† 𝒫 π‘œ ↔ 𝑆 βŠ† 𝒫 𝑂))
16 eleq12 2828 . . . . . 6 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ (π‘œ ∈ 𝑠 ↔ 𝑂 ∈ 𝑆))
17 simpr 486 . . . . . . 7 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ 𝑠 = 𝑆)
18 difeq1 4080 . . . . . . . . . 10 (π‘œ = 𝑂 β†’ (π‘œ βˆ– π‘₯) = (𝑂 βˆ– π‘₯))
1918adantr 482 . . . . . . . . 9 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ (π‘œ βˆ– π‘₯) = (𝑂 βˆ– π‘₯))
2019eleq1d 2823 . . . . . . . 8 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ ((π‘œ βˆ– π‘₯) ∈ 𝑠 ↔ (𝑂 βˆ– π‘₯) ∈ 𝑠))
21 eleq2 2827 . . . . . . . . 9 (𝑠 = 𝑆 β†’ ((𝑂 βˆ– π‘₯) ∈ 𝑠 ↔ (𝑂 βˆ– π‘₯) ∈ 𝑆))
2221adantl 483 . . . . . . . 8 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ ((𝑂 βˆ– π‘₯) ∈ 𝑠 ↔ (𝑂 βˆ– π‘₯) ∈ 𝑆))
2320, 22bitrd 279 . . . . . . 7 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ ((π‘œ βˆ– π‘₯) ∈ 𝑠 ↔ (𝑂 βˆ– π‘₯) ∈ 𝑆))
2417, 23raleqbidv 3322 . . . . . 6 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ (βˆ€π‘₯ ∈ 𝑠 (π‘œ βˆ– π‘₯) ∈ 𝑠 ↔ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆))
25 pweq 4579 . . . . . . . 8 (𝑠 = 𝑆 β†’ 𝒫 𝑠 = 𝒫 𝑆)
26 eleq2 2827 . . . . . . . . 9 (𝑠 = 𝑆 β†’ (βˆͺ π‘₯ ∈ 𝑠 ↔ βˆͺ π‘₯ ∈ 𝑆))
2726imbi2d 341 . . . . . . . 8 (𝑠 = 𝑆 β†’ ((π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑠) ↔ (π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆)))
2825, 27raleqbidv 3322 . . . . . . 7 (𝑠 = 𝑆 β†’ (βˆ€π‘₯ ∈ 𝒫 𝑠(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑠) ↔ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆)))
2928adantl 483 . . . . . 6 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ (βˆ€π‘₯ ∈ 𝒫 𝑠(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑠) ↔ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆)))
3016, 24, 293anbi123d 1437 . . . . 5 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ ((π‘œ ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (π‘œ βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑠)) ↔ (𝑂 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆))))
3115, 30anbi12d 632 . . . 4 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ ((𝑠 βŠ† 𝒫 π‘œ ∧ (π‘œ ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (π‘œ βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑠))) ↔ (𝑆 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆)))))
3210, 11, 31abfmpel 31613 . . 3 ((𝑂 ∈ V ∧ 𝑆 ∈ V) β†’ (𝑆 ∈ (sigAlgebraβ€˜π‘‚) ↔ (𝑆 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆)))))
3332a1i 11 . 2 (𝑆 ∈ V β†’ ((𝑂 ∈ V ∧ 𝑆 ∈ V) β†’ (𝑆 ∈ (sigAlgebraβ€˜π‘‚) ↔ (𝑆 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆))))))
344, 9, 33pm5.21ndd 381 1 (𝑆 ∈ V β†’ (𝑆 ∈ (sigAlgebraβ€˜π‘‚) ↔ (𝑆 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  Vcvv 3448   βˆ– cdif 3912   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870   class class class wbr 5110  β€˜cfv 6501  Ο‰com 7807   β‰Ό cdom 8888  sigAlgebracsiga 32747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-siga 32748
This theorem is referenced by:  baselsiga  32754  sigasspw  32755  issgon  32762  isrnsigau  32766  dmvlsiga  32768  pwsiga  32769  prsiga  32770  sigainb  32775  insiga  32776  sigapildsys  32801  imambfm  32902  carsgsiga  32962
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