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Theorem issiga 33098
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 6926 . . . 4 (𝑆 ∈ (sigAlgebraβ€˜π‘‚) β†’ 𝑂 ∈ V)
2 elex 3492 . . . 4 (𝑆 ∈ (sigAlgebraβ€˜π‘‚) β†’ 𝑆 ∈ V)
31, 2jca 512 . . 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 1194 . . . . 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 3492 . . . . 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 557 . 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 33095 . . . 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 33096 . . . 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 4615 . . . . . . 7 (π‘œ = 𝑂 β†’ 𝒫 π‘œ = 𝒫 𝑂)
1312sseq2d 4013 . . . . . 6 (π‘œ = 𝑂 β†’ (𝑠 βŠ† 𝒫 π‘œ ↔ 𝑠 βŠ† 𝒫 𝑂))
14 sseq1 4006 . . . . . 6 (𝑠 = 𝑆 β†’ (𝑠 βŠ† 𝒫 𝑂 ↔ 𝑆 βŠ† 𝒫 𝑂))
1513, 14sylan9bb 510 . . . . 5 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ (𝑠 βŠ† 𝒫 π‘œ ↔ 𝑆 βŠ† 𝒫 𝑂))
16 eleq12 2823 . . . . . 6 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ (π‘œ ∈ 𝑠 ↔ 𝑂 ∈ 𝑆))
17 simpr 485 . . . . . . 7 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ 𝑠 = 𝑆)
18 difeq1 4114 . . . . . . . . . 10 (π‘œ = 𝑂 β†’ (π‘œ βˆ– π‘₯) = (𝑂 βˆ– π‘₯))
1918adantr 481 . . . . . . . . 9 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ (π‘œ βˆ– π‘₯) = (𝑂 βˆ– π‘₯))
2019eleq1d 2818 . . . . . . . 8 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ ((π‘œ βˆ– π‘₯) ∈ 𝑠 ↔ (𝑂 βˆ– π‘₯) ∈ 𝑠))
21 eleq2 2822 . . . . . . . . 9 (𝑠 = 𝑆 β†’ ((𝑂 βˆ– π‘₯) ∈ 𝑠 ↔ (𝑂 βˆ– π‘₯) ∈ 𝑆))
2221adantl 482 . . . . . . . 8 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ ((𝑂 βˆ– π‘₯) ∈ 𝑠 ↔ (𝑂 βˆ– π‘₯) ∈ 𝑆))
2320, 22bitrd 278 . . . . . . 7 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ ((π‘œ βˆ– π‘₯) ∈ 𝑠 ↔ (𝑂 βˆ– π‘₯) ∈ 𝑆))
2417, 23raleqbidv 3342 . . . . . 6 ((π‘œ = 𝑂 ∧ 𝑠 = 𝑆) β†’ (βˆ€π‘₯ ∈ 𝑠 (π‘œ βˆ– π‘₯) ∈ 𝑠 ↔ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆))
25 pweq 4615 . . . . . . . 8 (𝑠 = 𝑆 β†’ 𝒫 𝑠 = 𝒫 𝑆)
26 eleq2 2822 . . . . . . . . 9 (𝑠 = 𝑆 β†’ (βˆͺ π‘₯ ∈ 𝑠 ↔ βˆͺ π‘₯ ∈ 𝑆))
2726imbi2d 340 . . . . . . . 8 (𝑠 = 𝑆 β†’ ((π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑠) ↔ (π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝑆)))
2825, 27raleqbidv 3342 . . . . . . 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 482 . . . . . 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 1436 . . . . 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 631 . . . 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 31867 . . 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 380 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆ– cdif 3944   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907   class class class wbr 5147  β€˜cfv 6540  Ο‰com 7851   β‰Ό cdom 8933  sigAlgebracsiga 33094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-siga 33095
This theorem is referenced by:  baselsiga  33101  sigasspw  33102  issgon  33109  isrnsigau  33113  dmvlsiga  33115  pwsiga  33116  prsiga  33117  sigainb  33122  insiga  33123  sigapildsys  33148  imambfm  33249  carsgsiga  33309
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