Theorem issiga 34108

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 6898	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
2		elex 3471	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
3	1, 2	jca 511	. . 3 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V))
4	3	a1i 11	. 2 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V)))
5		simpr1 1195	. . . . 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 17	. . . 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 556	. 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 34105	. . . 4 ⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
11		sigaex 34106	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V
12		pweq 4579	. . . . . . 7 ⊢ (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
13	12	sseq2d 3981	. . . . . 6 ⊢ (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑠 ⊆ 𝒫 𝑂))
14		sseq1 3974	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝒫 𝑂 ↔ 𝑆 ⊆ 𝒫 𝑂))
15	13, 14	sylan9bb 509	. . . . 5 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑆 ⊆ 𝒫 𝑂))
16		eleq12 2819	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑜 ∈ 𝑠 ↔ 𝑂 ∈ 𝑆))
17		simpr 484	. . . . . . 7 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
18		difeq1 4084	. . . . . . . . . 10 ⊢ (𝑜 = 𝑂 → (𝑜 ∖ 𝑥) = (𝑂 ∖ 𝑥))
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑜 ∖ 𝑥) = (𝑂 ∖ 𝑥))
20	19	eleq1d 2814	. . . . . . . 8 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑠))
21		eleq2 2818	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
22	21	adantl 481	. . . . . . . 8 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
23	20, 22	bitrd 279	. . . . . . 7 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
24	17, 23	raleqbidv 3321	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆))
25		pweq 4579	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
26		eleq2 2818	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∪ 𝑥 ∈ 𝑠 ↔ ∪ 𝑥 ∈ 𝑆))
27	26	imbi2d 340	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
28	25, 27	raleqbidv 3321	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
29	28	adantl 481	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
30	16, 24, 29	3anbi123d 1438	. . . . 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 632	. . . 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 32585	. . 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 379	1 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ∖ cdif 3913   ⊆ wss 3916  𝒫 cpw 4565  ∪ cuni 4873   class class class wbr 5109  ‘cfv 6513  ωcom 7844   ≼ cdom 8918  sigAlgebracsiga 34104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6515  df-fv 6521  df-siga 34105

This theorem is referenced by:  baselsiga  34111  sigasspw  34112  issgon  34119  isrnsigau  34123  dmvlsiga  34125  pwsiga  34126  prsiga  34127  sigainb  34132  insiga  34133  sigapildsys  34158  imambfm  34259  carsgsiga  34319