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Theorem issgon 30526
Description: Property of being a sigma-algebra with a given base set, noting that the base set of a sigma-algebra is actually its union set. (Contributed by Thierry Arnoux, 24-Sep-2016.) (Revised by Thierry Arnoux, 23-Oct-2016.)
Assertion
Ref Expression
issgon (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))

Proof of Theorem issgon
Dummy variables 𝑥 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvssunirn 6358 . . . 4 (sigAlgebra‘𝑂) ⊆ ran sigAlgebra
21sseli 3748 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ran sigAlgebra)
3 elex 3364 . . . 4 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
4 issiga 30514 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
5 elpwuni 4750 . . . . . . . 8 (𝑂𝑆 → (𝑆 ⊆ 𝒫 𝑂 𝑆 = 𝑂))
65biimpa 462 . . . . . . 7 ((𝑂𝑆𝑆 ⊆ 𝒫 𝑂) → 𝑆 = 𝑂)
7 ancom 452 . . . . . . 7 ((𝑆 ⊆ 𝒫 𝑂𝑂𝑆) ↔ (𝑂𝑆𝑆 ⊆ 𝒫 𝑂))
8 eqcom 2778 . . . . . . 7 (𝑂 = 𝑆 𝑆 = 𝑂)
96, 7, 83imtr4i 281 . . . . . 6 ((𝑆 ⊆ 𝒫 𝑂𝑂𝑆) → 𝑂 = 𝑆)
1093ad2antr1 1203 . . . . 5 ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂 = 𝑆)
114, 10syl6bi 243 . . . 4 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = 𝑆))
123, 11mpcom 38 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = 𝑆)
132, 12jca 501 . 2 (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))
14 elex 3364 . . . . 5 (𝑆 ran sigAlgebra → 𝑆 ∈ V)
15 isrnsiga 30516 . . . . . . . 8 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
1615simprbi 484 . . . . . . 7 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
17 elpwuni 4750 . . . . . . . . . . . . 13 (𝑜𝑆 → (𝑆 ⊆ 𝒫 𝑜 𝑆 = 𝑜))
1817biimpa 462 . . . . . . . . . . . 12 ((𝑜𝑆𝑆 ⊆ 𝒫 𝑜) → 𝑆 = 𝑜)
19 ancom 452 . . . . . . . . . . . 12 ((𝑆 ⊆ 𝒫 𝑜𝑜𝑆) ↔ (𝑜𝑆𝑆 ⊆ 𝒫 𝑜))
20 eqcom 2778 . . . . . . . . . . . 12 (𝑜 = 𝑆 𝑆 = 𝑜)
2118, 19, 203imtr4i 281 . . . . . . . . . . 11 ((𝑆 ⊆ 𝒫 𝑜𝑜𝑆) → 𝑜 = 𝑆)
22213ad2antr1 1203 . . . . . . . . . 10 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑜 = 𝑆)
23 pweq 4300 . . . . . . . . . . . 12 (𝑜 = 𝑆 → 𝒫 𝑜 = 𝒫 𝑆)
2423sseq2d 3782 . . . . . . . . . . 11 (𝑜 = 𝑆 → (𝑆 ⊆ 𝒫 𝑜𝑆 ⊆ 𝒫 𝑆))
25 eleq1 2838 . . . . . . . . . . . 12 (𝑜 = 𝑆 → (𝑜𝑆 𝑆𝑆))
26 difeq1 3872 . . . . . . . . . . . . . 14 (𝑜 = 𝑆 → (𝑜𝑥) = ( 𝑆𝑥))
2726eleq1d 2835 . . . . . . . . . . . . 13 (𝑜 = 𝑆 → ((𝑜𝑥) ∈ 𝑆 ↔ ( 𝑆𝑥) ∈ 𝑆))
2827ralbidv 3135 . . . . . . . . . . . 12 (𝑜 = 𝑆 → (∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ↔ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆))
2925, 283anbi12d 1548 . . . . . . . . . . 11 (𝑜 = 𝑆 → ((𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) ↔ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
3024, 29anbi12d 616 . . . . . . . . . 10 (𝑜 = 𝑆 → ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) ↔ (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
3122, 30syl 17 . . . . . . . . 9 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) ↔ (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
3231ibi 256 . . . . . . . 8 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
3332exlimiv 2010 . . . . . . 7 (∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
3416, 33syl 17 . . . . . 6 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
3534simprd 483 . . . . 5 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
3614, 35jca 501 . . . 4 (𝑆 ran sigAlgebra → (𝑆 ∈ V ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
37 eleq1 2838 . . . . . . . 8 (𝑂 = 𝑆 → (𝑂𝑆 𝑆𝑆))
38 difeq1 3872 . . . . . . . . . 10 (𝑂 = 𝑆 → (𝑂𝑥) = ( 𝑆𝑥))
3938eleq1d 2835 . . . . . . . . 9 (𝑂 = 𝑆 → ((𝑂𝑥) ∈ 𝑆 ↔ ( 𝑆𝑥) ∈ 𝑆))
4039ralbidv 3135 . . . . . . . 8 (𝑂 = 𝑆 → (∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ↔ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆))
4137, 403anbi12d 1548 . . . . . . 7 (𝑂 = 𝑆 → ((𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) ↔ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
4241biimprd 238 . . . . . 6 (𝑂 = 𝑆 → (( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) → (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
43 pwuni 4610 . . . . . . 7 𝑆 ⊆ 𝒫 𝑆
44 pweq 4300 . . . . . . 7 (𝑂 = 𝑆 → 𝒫 𝑂 = 𝒫 𝑆)
4543, 44syl5sseqr 3803 . . . . . 6 (𝑂 = 𝑆𝑆 ⊆ 𝒫 𝑂)
4642, 45jctild 515 . . . . 5 (𝑂 = 𝑆 → (( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) → (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
4746anim2d 599 . . . 4 (𝑂 = 𝑆 → ((𝑆 ∈ V ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → (𝑆 ∈ V ∧ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))))
484biimpar 463 . . . 4 ((𝑆 ∈ V ∧ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))) → 𝑆 ∈ (sigAlgebra‘𝑂))
4936, 47, 48syl56 36 . . 3 (𝑂 = 𝑆 → (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘𝑂)))
5049impcom 394 . 2 ((𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆) → 𝑆 ∈ (sigAlgebra‘𝑂))
5113, 50impbii 199 1 (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wral 3061  Vcvv 3351  cdif 3720  wss 3723  𝒫 cpw 4297   cuni 4574   class class class wbr 4786  ran crn 5250  cfv 6031  ωcom 7212  cdom 8107  sigAlgebracsiga 30510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-siga 30511
This theorem is referenced by:  sgon  30527  unisg  30546  sxsigon  30595  sxuni  30596  1stmbfm  30662  2ndmbfm  30663  mbfmvolf  30668
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