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Theorem issgon 34125
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 6938 . . . 4 (sigAlgebra‘𝑂) ⊆ ran sigAlgebra
21sseli 3978 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ran sigAlgebra)
3 elex 3500 . . . 4 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
4 issiga 34114 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
5 elpwuni 5104 . . . . . . . 8 (𝑂𝑆 → (𝑆 ⊆ 𝒫 𝑂 𝑆 = 𝑂))
65biimpa 476 . . . . . . 7 ((𝑂𝑆𝑆 ⊆ 𝒫 𝑂) → 𝑆 = 𝑂)
7 ancom 460 . . . . . . 7 ((𝑆 ⊆ 𝒫 𝑂𝑂𝑆) ↔ (𝑂𝑆𝑆 ⊆ 𝒫 𝑂))
8 eqcom 2743 . . . . . . 7 (𝑂 = 𝑆 𝑆 = 𝑂)
96, 7, 83imtr4i 292 . . . . . 6 ((𝑆 ⊆ 𝒫 𝑂𝑂𝑆) → 𝑂 = 𝑆)
1093ad2antr1 1188 . . . . 5 ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂 = 𝑆)
114, 10biimtrdi 253 . . . 4 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = 𝑆))
123, 11mpcom 38 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = 𝑆)
132, 12jca 511 . 2 (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))
14 elex 3500 . . . . 5 (𝑆 ran sigAlgebra → 𝑆 ∈ V)
15 isrnsiga 34115 . . . . . . . 8 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
1615simprbi 496 . . . . . . 7 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
17 elpwuni 5104 . . . . . . . . . . . . 13 (𝑜𝑆 → (𝑆 ⊆ 𝒫 𝑜 𝑆 = 𝑜))
1817biimpa 476 . . . . . . . . . . . 12 ((𝑜𝑆𝑆 ⊆ 𝒫 𝑜) → 𝑆 = 𝑜)
19 ancom 460 . . . . . . . . . . . 12 ((𝑆 ⊆ 𝒫 𝑜𝑜𝑆) ↔ (𝑜𝑆𝑆 ⊆ 𝒫 𝑜))
20 eqcom 2743 . . . . . . . . . . . 12 (𝑜 = 𝑆 𝑆 = 𝑜)
2118, 19, 203imtr4i 292 . . . . . . . . . . 11 ((𝑆 ⊆ 𝒫 𝑜𝑜𝑆) → 𝑜 = 𝑆)
22213ad2antr1 1188 . . . . . . . . . 10 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑜 = 𝑆)
23 pweq 4613 . . . . . . . . . . . 12 (𝑜 = 𝑆 → 𝒫 𝑜 = 𝒫 𝑆)
2423sseq2d 4015 . . . . . . . . . . 11 (𝑜 = 𝑆 → (𝑆 ⊆ 𝒫 𝑜𝑆 ⊆ 𝒫 𝑆))
25 eleq1 2828 . . . . . . . . . . . 12 (𝑜 = 𝑆 → (𝑜𝑆 𝑆𝑆))
26 difeq1 4118 . . . . . . . . . . . . . 14 (𝑜 = 𝑆 → (𝑜𝑥) = ( 𝑆𝑥))
2726eleq1d 2825 . . . . . . . . . . . . 13 (𝑜 = 𝑆 → ((𝑜𝑥) ∈ 𝑆 ↔ ( 𝑆𝑥) ∈ 𝑆))
2827ralbidv 3177 . . . . . . . . . . . 12 (𝑜 = 𝑆 → (∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ↔ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆))
2925, 283anbi12d 1438 . . . . . . . . . . 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 632 . . . . . . . . . 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 267 . . . . . . . 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 1929 . . . . . . 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 495 . . . . 5 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
3614, 35jca 511 . . . 4 (𝑆 ran sigAlgebra → (𝑆 ∈ V ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
37 eleq1 2828 . . . . . . . 8 (𝑂 = 𝑆 → (𝑂𝑆 𝑆𝑆))
38 difeq1 4118 . . . . . . . . . 10 (𝑂 = 𝑆 → (𝑂𝑥) = ( 𝑆𝑥))
3938eleq1d 2825 . . . . . . . . 9 (𝑂 = 𝑆 → ((𝑂𝑥) ∈ 𝑆 ↔ ( 𝑆𝑥) ∈ 𝑆))
4039ralbidv 3177 . . . . . . . 8 (𝑂 = 𝑆 → (∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ↔ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆))
4137, 403anbi12d 1438 . . . . . . 7 (𝑂 = 𝑆 → ((𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) ↔ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
4241biimprd 248 . . . . . 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 4944 . . . . . . 7 𝑆 ⊆ 𝒫 𝑆
44 pweq 4613 . . . . . . 7 (𝑂 = 𝑆 → 𝒫 𝑂 = 𝒫 𝑆)
4543, 44sseqtrrid 4026 . . . . . 6 (𝑂 = 𝑆𝑆 ⊆ 𝒫 𝑂)
4642, 45jctild 525 . . . . 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 612 . . . 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 477 . . . 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 407 . 2 ((𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆) → 𝑆 ∈ (sigAlgebra‘𝑂))
5113, 50impbii 209 1 (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  wral 3060  Vcvv 3479  cdif 3947  wss 3950  𝒫 cpw 4599   cuni 4906   class class class wbr 5142  ran crn 5685  cfv 6560  ωcom 7888  cdom 8984  sigAlgebracsiga 34110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-siga 34111
This theorem is referenced by:  sgon  34126  unisg  34145  sxsigon  34194  sxuni  34195  1stmbfm  34263  2ndmbfm  34264  mbfmvolf  34269
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