Theorem issiga 34744

Description: An alternative definition of the sigma-algebra, for a given base set. (Contributed by Thierry Arnoux, 19-Sep-2016.)

Assertion

Ref	Expression
issiga	⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))

Distinct variable groups:   𝑥,𝑂   𝑥,𝑆

Proof of Theorem issiga

Dummy variables 𝑜 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6920	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
2		elex 3471	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
3	1, 2	jca 521	. . 3 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V))
4	3	a1i 11	. 2 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V)))
5		simpr1 1213	. . . . 5 ⊢ ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → 𝑂 ∈ 𝑆)
6		elex 3471	. . . . 5 ⊢ (𝑂 ∈ 𝑆 → 𝑂 ∈ V)
7	5, 6	syl 18	. . . 4 ⊢ ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → 𝑂 ∈ V)
8	7	a1i 11	. . 3 ⊢ (𝑆 ∈ V → ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → 𝑂 ∈ V))
9	8	anc2ri 566	. 2 ⊢ (𝑆 ∈ V → ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → (𝑂 ∈ V ∧ 𝑆 ∈ V)))
10		df-siga 34741	. . . 4 ⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
11		sigaex 34742	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V
12		pweq 4570	. . . . . . 7 ⊢ (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
13	12	sseq2d 3940	. . . . . 6 ⊢ (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑠 ⊆ 𝒫 𝑂))
14		sseq1 3933	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝒫 𝑂 ↔ 𝑆 ⊆ 𝒫 𝑂))
15	13, 14	sylan9bb 519	. . . . 5 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑆 ⊆ 𝒫 𝑂))
16		eleq12 2850	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑜 ∈ 𝑠 ↔ 𝑂 ∈ 𝑆))
17		simpr 490	. . . . . . 7 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
18		difeq1 4056	. . . . . . . . . 10 ⊢ (𝑜 = 𝑂 → (𝑜 ∖ 𝑥) = (𝑂 ∖ 𝑥))
19	18	adantr 486	. . . . . . . . 9 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑜 ∖ 𝑥) = (𝑂 ∖ 𝑥))
20	19	eleq1d 2845	. . . . . . . 8 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑠))
21		eleq2 2849	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
22	21	adantl 487	. . . . . . . 8 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
23	20, 22	bitrd 282	. . . . . . 7 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
24	17, 23	raleqbidv 3334	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆))
25		pweq 4570	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
26		eleq2 2849	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∪ 𝑥 ∈ 𝑠 ↔ ∪ 𝑥 ∈ 𝑆))
27	26	imbi2d 343	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
28	25, 27	raleqbidv 3334	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
29	28	adantl 487	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
30	16, 24, 29	3anbi123d 1464	. . . . 5 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)) ↔ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
31	15, 30	anbi12d 644	. . . 4 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
32	10, 11, 31	abfmpel 33249	. . 3 ⊢ ((𝑂 ∈ V ∧ 𝑆 ∈ V) → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
33	32	a1i 11	. 2 ⊢ (𝑆 ∈ V → ((𝑂 ∈ V ∧ 𝑆 ∈ V) → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))))
34	4, 9, 33	pm5.21ndd 382	1 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045  𝒫 cpw 4556  ∪ cuni 4866   class class class wbr 5102  ‘cfv 6538  ωcom 7877   ≼ cdom 8971  sigAlgebracsiga 34740

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-siga 34741

This theorem is used by:  baselsiga  34747  sigasspw  34748  issgon  34755  isrnsigau  34759  dmvlsiga  34761  pwsiga  34762  prsiga  34763  sigainb  34769  insiga  34770  sigapildsys  34795  imambfm  34894  carsgsiga  34954