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Theorem issgon 29513
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 6127 . . . 4 (sigAlgebra‘𝑂) ⊆ ran sigAlgebra
21sseli 3564 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ran sigAlgebra)
3 elex 3185 . . . 4 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
4 issiga 29501 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
5 elpwuni 4549 . . . . . . . 8 (𝑂𝑆 → (𝑆 ⊆ 𝒫 𝑂 𝑆 = 𝑂))
65biimpa 500 . . . . . . 7 ((𝑂𝑆𝑆 ⊆ 𝒫 𝑂) → 𝑆 = 𝑂)
7 ancom 465 . . . . . . 7 ((𝑆 ⊆ 𝒫 𝑂𝑂𝑆) ↔ (𝑂𝑆𝑆 ⊆ 𝒫 𝑂))
8 eqcom 2617 . . . . . . 7 (𝑂 = 𝑆 𝑆 = 𝑂)
96, 7, 83imtr4i 280 . . . . . 6 ((𝑆 ⊆ 𝒫 𝑂𝑂𝑆) → 𝑂 = 𝑆)
1093ad2antr1 1219 . . . . 5 ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂 = 𝑆)
114, 10syl6bi 242 . . . 4 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = 𝑆))
123, 11mpcom 37 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = 𝑆)
132, 12jca 553 . 2 (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))
14 elex 3185 . . . . 5 (𝑆 ran sigAlgebra → 𝑆 ∈ V)
15 isrnsiga 29503 . . . . . . . 8 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
1615simprbi 479 . . . . . . 7 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
17 elpwuni 4549 . . . . . . . . . . . . 13 (𝑜𝑆 → (𝑆 ⊆ 𝒫 𝑜 𝑆 = 𝑜))
1817biimpa 500 . . . . . . . . . . . 12 ((𝑜𝑆𝑆 ⊆ 𝒫 𝑜) → 𝑆 = 𝑜)
19 ancom 465 . . . . . . . . . . . 12 ((𝑆 ⊆ 𝒫 𝑜𝑜𝑆) ↔ (𝑜𝑆𝑆 ⊆ 𝒫 𝑜))
20 eqcom 2617 . . . . . . . . . . . 12 (𝑜 = 𝑆 𝑆 = 𝑜)
2118, 19, 203imtr4i 280 . . . . . . . . . . 11 ((𝑆 ⊆ 𝒫 𝑜𝑜𝑆) → 𝑜 = 𝑆)
22213ad2antr1 1219 . . . . . . . . . 10 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑜 = 𝑆)
23 pweq 4111 . . . . . . . . . . . 12 (𝑜 = 𝑆 → 𝒫 𝑜 = 𝒫 𝑆)
2423sseq2d 3596 . . . . . . . . . . 11 (𝑜 = 𝑆 → (𝑆 ⊆ 𝒫 𝑜𝑆 ⊆ 𝒫 𝑆))
25 eleq1 2676 . . . . . . . . . . . 12 (𝑜 = 𝑆 → (𝑜𝑆 𝑆𝑆))
26 difeq1 3683 . . . . . . . . . . . . . 14 (𝑜 = 𝑆 → (𝑜𝑥) = ( 𝑆𝑥))
2726eleq1d 2672 . . . . . . . . . . . . 13 (𝑜 = 𝑆 → ((𝑜𝑥) ∈ 𝑆 ↔ ( 𝑆𝑥) ∈ 𝑆))
2827ralbidv 2969 . . . . . . . . . . . 12 (𝑜 = 𝑆 → (∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ↔ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆))
2925, 283anbi12d 1392 . . . . . . . . . . 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 743 . . . . . . . . . 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 255 . . . . . . . 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 1845 . . . . . . 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 478 . . . . 5 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
3614, 35jca 553 . . . 4 (𝑆 ran sigAlgebra → (𝑆 ∈ V ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
37 eleq1 2676 . . . . . . . 8 (𝑂 = 𝑆 → (𝑂𝑆 𝑆𝑆))
38 difeq1 3683 . . . . . . . . . 10 (𝑂 = 𝑆 → (𝑂𝑥) = ( 𝑆𝑥))
3938eleq1d 2672 . . . . . . . . 9 (𝑂 = 𝑆 → ((𝑂𝑥) ∈ 𝑆 ↔ ( 𝑆𝑥) ∈ 𝑆))
4039ralbidv 2969 . . . . . . . 8 (𝑂 = 𝑆 → (∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ↔ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆))
4137, 403anbi12d 1392 . . . . . . 7 (𝑂 = 𝑆 → ((𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) ↔ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
4241biimprd 237 . . . . . 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 4825 . . . . . . 7 𝑆 ⊆ 𝒫 𝑆
44 pweq 4111 . . . . . . 7 (𝑂 = 𝑆 → 𝒫 𝑂 = 𝒫 𝑆)
4543, 44syl5sseqr 3617 . . . . . 6 (𝑂 = 𝑆𝑆 ⊆ 𝒫 𝑂)
4642, 45jctild 564 . . . . 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 587 . . . 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 501 . . . 4 ((𝑆 ∈ V ∧ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))) → 𝑆 ∈ (sigAlgebra‘𝑂))
4936, 47, 48syl56 35 . . 3 (𝑂 = 𝑆 → (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘𝑂)))
5049impcom 445 . 2 ((𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆) → 𝑆 ∈ (sigAlgebra‘𝑂))
5113, 50impbii 198 1 (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  wral 2896  Vcvv 3173  cdif 3537  wss 3540  𝒫 cpw 4108   cuni 4372   class class class wbr 4583  ran crn 5039  cfv 5804  ωcom 6957  cdom 7839  sigAlgebracsiga 29497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-siga 29498
This theorem is referenced by:  sgon  29514  unisg  29533  sxsigon  29582  sxuni  29583  1stmbfm  29649  2ndmbfm  29650  mbfmvolf  29655
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