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Theorem issgon 31281
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 6692 . . . 4 (sigAlgebra‘𝑂) ⊆ ran sigAlgebra
21sseli 3960 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ran sigAlgebra)
3 elex 3510 . . . 4 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
4 issiga 31270 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
5 elpwuni 5018 . . . . . . . 8 (𝑂𝑆 → (𝑆 ⊆ 𝒫 𝑂 𝑆 = 𝑂))
65biimpa 477 . . . . . . 7 ((𝑂𝑆𝑆 ⊆ 𝒫 𝑂) → 𝑆 = 𝑂)
7 ancom 461 . . . . . . 7 ((𝑆 ⊆ 𝒫 𝑂𝑂𝑆) ↔ (𝑂𝑆𝑆 ⊆ 𝒫 𝑂))
8 eqcom 2825 . . . . . . 7 (𝑂 = 𝑆 𝑆 = 𝑂)
96, 7, 83imtr4i 293 . . . . . 6 ((𝑆 ⊆ 𝒫 𝑂𝑂𝑆) → 𝑂 = 𝑆)
1093ad2antr1 1180 . . . . 5 ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂 = 𝑆)
114, 10syl6bi 254 . . . 4 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = 𝑆))
123, 11mpcom 38 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = 𝑆)
132, 12jca 512 . 2 (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))
14 elex 3510 . . . . 5 (𝑆 ran sigAlgebra → 𝑆 ∈ V)
15 isrnsiga 31271 . . . . . . . 8 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
1615simprbi 497 . . . . . . 7 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
17 elpwuni 5018 . . . . . . . . . . . . 13 (𝑜𝑆 → (𝑆 ⊆ 𝒫 𝑜 𝑆 = 𝑜))
1817biimpa 477 . . . . . . . . . . . 12 ((𝑜𝑆𝑆 ⊆ 𝒫 𝑜) → 𝑆 = 𝑜)
19 ancom 461 . . . . . . . . . . . 12 ((𝑆 ⊆ 𝒫 𝑜𝑜𝑆) ↔ (𝑜𝑆𝑆 ⊆ 𝒫 𝑜))
20 eqcom 2825 . . . . . . . . . . . 12 (𝑜 = 𝑆 𝑆 = 𝑜)
2118, 19, 203imtr4i 293 . . . . . . . . . . 11 ((𝑆 ⊆ 𝒫 𝑜𝑜𝑆) → 𝑜 = 𝑆)
22213ad2antr1 1180 . . . . . . . . . 10 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑜 = 𝑆)
23 pweq 4538 . . . . . . . . . . . 12 (𝑜 = 𝑆 → 𝒫 𝑜 = 𝒫 𝑆)
2423sseq2d 3996 . . . . . . . . . . 11 (𝑜 = 𝑆 → (𝑆 ⊆ 𝒫 𝑜𝑆 ⊆ 𝒫 𝑆))
25 eleq1 2897 . . . . . . . . . . . 12 (𝑜 = 𝑆 → (𝑜𝑆 𝑆𝑆))
26 difeq1 4089 . . . . . . . . . . . . . 14 (𝑜 = 𝑆 → (𝑜𝑥) = ( 𝑆𝑥))
2726eleq1d 2894 . . . . . . . . . . . . 13 (𝑜 = 𝑆 → ((𝑜𝑥) ∈ 𝑆 ↔ ( 𝑆𝑥) ∈ 𝑆))
2827ralbidv 3194 . . . . . . . . . . . 12 (𝑜 = 𝑆 → (∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ↔ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆))
2925, 283anbi12d 1428 . . . . . . . . . . 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 630 . . . . . . . . . 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 268 . . . . . . . 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 1922 . . . . . . 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 496 . . . . 5 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
3614, 35jca 512 . . . 4 (𝑆 ran sigAlgebra → (𝑆 ∈ V ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
37 eleq1 2897 . . . . . . . 8 (𝑂 = 𝑆 → (𝑂𝑆 𝑆𝑆))
38 difeq1 4089 . . . . . . . . . 10 (𝑂 = 𝑆 → (𝑂𝑥) = ( 𝑆𝑥))
3938eleq1d 2894 . . . . . . . . 9 (𝑂 = 𝑆 → ((𝑂𝑥) ∈ 𝑆 ↔ ( 𝑆𝑥) ∈ 𝑆))
4039ralbidv 3194 . . . . . . . 8 (𝑂 = 𝑆 → (∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ↔ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆))
4137, 403anbi12d 1428 . . . . . . 7 (𝑂 = 𝑆 → ((𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) ↔ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
4241biimprd 249 . . . . . 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 4866 . . . . . . 7 𝑆 ⊆ 𝒫 𝑆
44 pweq 4538 . . . . . . 7 (𝑂 = 𝑆 → 𝒫 𝑂 = 𝒫 𝑆)
4543, 44sseqtrrid 4017 . . . . . 6 (𝑂 = 𝑆𝑆 ⊆ 𝒫 𝑂)
4642, 45jctild 526 . . . . 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 611 . . . 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 478 . . . 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 408 . 2 ((𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆) → 𝑆 ∈ (sigAlgebra‘𝑂))
5113, 50impbii 210 1 (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wral 3135  Vcvv 3492  cdif 3930  wss 3933  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  ran crn 5549  cfv 6348  ωcom 7569  cdom 8495  sigAlgebracsiga 31266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-siga 31267
This theorem is referenced by:  sgon  31282  unisg  31301  sxsigon  31350  sxuni  31351  1stmbfm  31417  2ndmbfm  31418  mbfmvolf  31423
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