Theorem issgon 34096

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.

Step	Hyp	Ref	Expression
1		fvssunirn 6853	. . . 4 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra
2	1	sseli 3931	. . 3 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra)
3		elex 3457	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
4		issiga 34085	. . . . 5 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
5		elpwuni 5054	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑆 → (𝑆 ⊆ 𝒫 𝑂 ↔ ∪ 𝑆 = 𝑂))
6	5	biimpa 476	. . . . . . 7 ⊢ ((𝑂 ∈ 𝑆 ∧ 𝑆 ⊆ 𝒫 𝑂) → ∪ 𝑆 = 𝑂)
7		ancom 460	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝒫 𝑂 ∧ 𝑂 ∈ 𝑆) ↔ (𝑂 ∈ 𝑆 ∧ 𝑆 ⊆ 𝒫 𝑂))
8		eqcom 2736	. . . . . . 7 ⊢ (𝑂 = ∪ 𝑆 ↔ ∪ 𝑆 = 𝑂)
9	6, 7, 8	3imtr4i 292	. . . . . 6 ⊢ ((𝑆 ⊆ 𝒫 𝑂 ∧ 𝑂 ∈ 𝑆) → 𝑂 = ∪ 𝑆)
10	9	3ad2antr1 1189	. . . . 5 ⊢ ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → 𝑂 = ∪ 𝑆)
11	4, 10	biimtrdi 253	. . . 4 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = ∪ 𝑆))
12	3, 11	mpcom 38	. . 3 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 = ∪ 𝑆)
13	2, 12	jca 511	. 2 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑂 = ∪ 𝑆))
14		elex 3457	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ V)
15		isrnsiga 34086	. . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
16	15	simprbi 496	. . . . . . 7 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
17		elpwuni 5054	. . . . . . . . . . . . 13 ⊢ (𝑜 ∈ 𝑆 → (𝑆 ⊆ 𝒫 𝑜 ↔ ∪ 𝑆 = 𝑜))
18	17	biimpa 476	. . . . . . . . . . . 12 ⊢ ((𝑜 ∈ 𝑆 ∧ 𝑆 ⊆ 𝒫 𝑜) → ∪ 𝑆 = 𝑜)
19		ancom 460	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ 𝒫 𝑜 ∧ 𝑜 ∈ 𝑆) ↔ (𝑜 ∈ 𝑆 ∧ 𝑆 ⊆ 𝒫 𝑜))
20		eqcom 2736	. . . . . . . . . . . 12 ⊢ (𝑜 = ∪ 𝑆 ↔ ∪ 𝑆 = 𝑜)
21	18, 19, 20	3imtr4i 292	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ 𝒫 𝑜 ∧ 𝑜 ∈ 𝑆) → 𝑜 = ∪ 𝑆)
22	21	3ad2antr1 1189	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → 𝑜 = ∪ 𝑆)
23		pweq 4565	. . . . . . . . . . . 12 ⊢ (𝑜 = ∪ 𝑆 → 𝒫 𝑜 = 𝒫 ∪ 𝑆)
24	23	sseq2d 3968	. . . . . . . . . . 11 ⊢ (𝑜 = ∪ 𝑆 → (𝑆 ⊆ 𝒫 𝑜 ↔ 𝑆 ⊆ 𝒫 ∪ 𝑆))
25		eleq1 2816	. . . . . . . . . . . 12 ⊢ (𝑜 = ∪ 𝑆 → (𝑜 ∈ 𝑆 ↔ ∪ 𝑆 ∈ 𝑆))
26		difeq1 4070	. . . . . . . . . . . . . 14 ⊢ (𝑜 = ∪ 𝑆 → (𝑜 ∖ 𝑥) = (∪ 𝑆 ∖ 𝑥))
27	26	eleq1d 2813	. . . . . . . . . . . . 13 ⊢ (𝑜 = ∪ 𝑆 → ((𝑜 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝑥) ∈ 𝑆))
28	27	ralbidv 3152	. . . . . . . . . . . 12 ⊢ (𝑜 = ∪ 𝑆 → (∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆))
29	25, 28	3anbi12d 1439	. . . . . . . . . . 11 ⊢ (𝑜 = ∪ 𝑆 → ((𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)) ↔ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
30	24, 29	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑜 = ∪ 𝑆 → ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) ↔ (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
31	22, 30	syl 17	. . . . . . . . 9 ⊢ ((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))) ↔ (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
32	31	ibi 267	. . . . . . . 8 ⊢ ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
33	32	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
34	16, 33	syl 17	. . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
35	34	simprd 495	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
36	14, 35	jca 511	. . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ∈ V ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
37		eleq1 2816	. . . . . . . 8 ⊢ (𝑂 = ∪ 𝑆 → (𝑂 ∈ 𝑆 ↔ ∪ 𝑆 ∈ 𝑆))
38		difeq1 4070	. . . . . . . . . 10 ⊢ (𝑂 = ∪ 𝑆 → (𝑂 ∖ 𝑥) = (∪ 𝑆 ∖ 𝑥))
39	38	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑂 = ∪ 𝑆 → ((𝑂 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝑥) ∈ 𝑆))
40	39	ralbidv 3152	. . . . . . . 8 ⊢ (𝑂 = ∪ 𝑆 → (∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆))
41	37, 40	3anbi12d 1439	. . . . . . 7 ⊢ (𝑂 = ∪ 𝑆 → ((𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)) ↔ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
42	41	biimprd 248	. . . . . 6 ⊢ (𝑂 = ∪ 𝑆 → ((∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)) → (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
43		pwuni 4895	. . . . . . 7 ⊢ 𝑆 ⊆ 𝒫 ∪ 𝑆
44		pweq 4565	. . . . . . 7 ⊢ (𝑂 = ∪ 𝑆 → 𝒫 𝑂 = 𝒫 ∪ 𝑆)
45	43, 44	sseqtrrid 3979	. . . . . 6 ⊢ (𝑂 = ∪ 𝑆 → 𝑆 ⊆ 𝒫 𝑂)
46	42, 45	jctild 525	. . . . 5 ⊢ (𝑂 = ∪ 𝑆 → ((∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 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)))))
47	46	anim2d 612	. . . 4 ⊢ (𝑂 = ∪ 𝑆 → ((𝑆 ∈ V ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → (𝑆 ∈ V ∧ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))))
48	4	biimpar 477	. . . 4 ⊢ ((𝑆 ∈ V ∧ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))) → 𝑆 ∈ (sigAlgebra‘𝑂))
49	36, 47, 48	syl56 36	. . 3 ⊢ (𝑂 = ∪ 𝑆 → (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘𝑂)))
50	49	impcom 407	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑂 = ∪ 𝑆) → 𝑆 ∈ (sigAlgebra‘𝑂))
51	13, 50	impbii 209	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑂 = ∪ 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858   class class class wbr 5092  ran crn 5620  ‘cfv 6482  ωcom 7799   ≼ cdom 8870  sigAlgebracsiga 34081

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 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490  df-siga 34082

This theorem is referenced by:  sgon  34097  unisg  34116  sxsigon  34165  sxuni  34166  1stmbfm  34234  2ndmbfm  34235  mbfmvolf  34240