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Theorem issgon 30511
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 6437 . . . 4 (sigAlgebra‘𝑂) ⊆ ran sigAlgebra
21sseli 3794 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ran sigAlgebra)
3 elex 3406 . . . 4 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
4 issiga 30499 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
5 elpwuni 4808 . . . . . . . 8 (𝑂𝑆 → (𝑆 ⊆ 𝒫 𝑂 𝑆 = 𝑂))
65biimpa 464 . . . . . . 7 ((𝑂𝑆𝑆 ⊆ 𝒫 𝑂) → 𝑆 = 𝑂)
7 ancom 450 . . . . . . 7 ((𝑆 ⊆ 𝒫 𝑂𝑂𝑆) ↔ (𝑂𝑆𝑆 ⊆ 𝒫 𝑂))
8 eqcom 2813 . . . . . . 7 (𝑂 = 𝑆 𝑆 = 𝑂)
96, 7, 83imtr4i 283 . . . . . 6 ((𝑆 ⊆ 𝒫 𝑂𝑂𝑆) → 𝑂 = 𝑆)
1093ad2antr1 1232 . . . . 5 ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂 = 𝑆)
114, 10syl6bi 244 . . . 4 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = 𝑆))
123, 11mpcom 38 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = 𝑆)
132, 12jca 503 . 2 (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))
14 elex 3406 . . . . 5 (𝑆 ran sigAlgebra → 𝑆 ∈ V)
15 isrnsiga 30501 . . . . . . . 8 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
1615simprbi 486 . . . . . . 7 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
17 elpwuni 4808 . . . . . . . . . . . . 13 (𝑜𝑆 → (𝑆 ⊆ 𝒫 𝑜 𝑆 = 𝑜))
1817biimpa 464 . . . . . . . . . . . 12 ((𝑜𝑆𝑆 ⊆ 𝒫 𝑜) → 𝑆 = 𝑜)
19 ancom 450 . . . . . . . . . . . 12 ((𝑆 ⊆ 𝒫 𝑜𝑜𝑆) ↔ (𝑜𝑆𝑆 ⊆ 𝒫 𝑜))
20 eqcom 2813 . . . . . . . . . . . 12 (𝑜 = 𝑆 𝑆 = 𝑜)
2118, 19, 203imtr4i 283 . . . . . . . . . . 11 ((𝑆 ⊆ 𝒫 𝑜𝑜𝑆) → 𝑜 = 𝑆)
22213ad2antr1 1232 . . . . . . . . . 10 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑜 = 𝑆)
23 pweq 4354 . . . . . . . . . . . 12 (𝑜 = 𝑆 → 𝒫 𝑜 = 𝒫 𝑆)
2423sseq2d 3830 . . . . . . . . . . 11 (𝑜 = 𝑆 → (𝑆 ⊆ 𝒫 𝑜𝑆 ⊆ 𝒫 𝑆))
25 eleq1 2873 . . . . . . . . . . . 12 (𝑜 = 𝑆 → (𝑜𝑆 𝑆𝑆))
26 difeq1 3920 . . . . . . . . . . . . . 14 (𝑜 = 𝑆 → (𝑜𝑥) = ( 𝑆𝑥))
2726eleq1d 2870 . . . . . . . . . . . . 13 (𝑜 = 𝑆 → ((𝑜𝑥) ∈ 𝑆 ↔ ( 𝑆𝑥) ∈ 𝑆))
2827ralbidv 3174 . . . . . . . . . . . 12 (𝑜 = 𝑆 → (∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ↔ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆))
2925, 283anbi12d 1554 . . . . . . . . . . 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 618 . . . . . . . . . 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 258 . . . . . . . 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 2021 . . . . . . 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 485 . . . . 5 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
3614, 35jca 503 . . . 4 (𝑆 ran sigAlgebra → (𝑆 ∈ V ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
37 eleq1 2873 . . . . . . . 8 (𝑂 = 𝑆 → (𝑂𝑆 𝑆𝑆))
38 difeq1 3920 . . . . . . . . . 10 (𝑂 = 𝑆 → (𝑂𝑥) = ( 𝑆𝑥))
3938eleq1d 2870 . . . . . . . . 9 (𝑂 = 𝑆 → ((𝑂𝑥) ∈ 𝑆 ↔ ( 𝑆𝑥) ∈ 𝑆))
4039ralbidv 3174 . . . . . . . 8 (𝑂 = 𝑆 → (∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ↔ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆))
4137, 403anbi12d 1554 . . . . . . 7 (𝑂 = 𝑆 → ((𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) ↔ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
4241biimprd 239 . . . . . 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 4668 . . . . . . 7 𝑆 ⊆ 𝒫 𝑆
44 pweq 4354 . . . . . . 7 (𝑂 = 𝑆 → 𝒫 𝑂 = 𝒫 𝑆)
4543, 44syl5sseqr 3851 . . . . . 6 (𝑂 = 𝑆𝑆 ⊆ 𝒫 𝑂)
4642, 45jctild 517 . . . . 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 601 . . . 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 465 . . . 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 396 . 2 ((𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆) → 𝑆 ∈ (sigAlgebra‘𝑂))
5113, 50impbii 200 1 (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2156  wral 3096  Vcvv 3391  cdif 3766  wss 3769  𝒫 cpw 4351   cuni 4630   class class class wbr 4844  ran crn 5312  cfv 6101  ωcom 7295  cdom 8190  sigAlgebracsiga 30495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109  df-siga 30496
This theorem is referenced by:  sgon  30512  unisg  30531  sxsigon  30580  sxuni  30581  1stmbfm  30647  2ndmbfm  30648  mbfmvolf  30653
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