Theorem dfsalgen2 46337

Description: Alternate characterization of the sigma-algebra generated by a set. It is the smallest sigma-algebra, on the same base set, that includes the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypothesis

Ref	Expression
dfsalgen2.1	⊢ (𝜑 → 𝑋 ∈ 𝑉)

Assertion

Ref	Expression
dfsalgen2	⊢ (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))))

Distinct variable groups:   𝑦,𝑆   𝑦,𝑋   𝜑,𝑦

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem dfsalgen2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . 8 ⊢ ((SalGen‘𝑋) = 𝑆 → (SalGen‘𝑋) = 𝑆)
2	1	eqcomd 2742	. . . . . . 7 ⊢ ((SalGen‘𝑋) = 𝑆 → 𝑆 = (SalGen‘𝑋))
3	2	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = (SalGen‘𝑋))
4		dfsalgen2.1	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
5		salgencl 46328	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg)
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg)
7	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) ∈ SAlg)
8	3, 7	eqeltrd 2835	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 ∈ SAlg)
9		unieq 4899	. . . . . . 7 ⊢ ((SalGen‘𝑋) = 𝑆 → ∪ (SalGen‘𝑋) = ∪ 𝑆)
10	9	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ (SalGen‘𝑋) = ∪ 𝑆)
11	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ∈ 𝑉)
12		eqid 2736	. . . . . . 7 ⊢ (SalGen‘𝑋) = (SalGen‘𝑋)
13		eqid 2736	. . . . . . 7 ⊢ ∪ 𝑋 = ∪ 𝑋
14	11, 12, 13	salgenuni 46333	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ (SalGen‘𝑋) = ∪ 𝑋)
15	10, 14	eqtr3d 2773	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ 𝑆 = ∪ 𝑋)
16	12	sssalgen 46331	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (SalGen‘𝑋))
17	11, 16	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ (SalGen‘𝑋))
18		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
19	17, 18	sseqtrd 4000	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ 𝑆)
20	8, 15, 19	3jca 1128	. . . 4 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))
21	3	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑆 = (SalGen‘𝑋))
22	21	adantrl 716	. . . . . . 7 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 = (SalGen‘𝑋))
23	11	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ∈ 𝑉)
24	23	adantrl 716	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑋 ∈ 𝑉)
25		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ SAlg)
26	25	adantrl 716	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑦 ∈ SAlg)
27		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ⊆ 𝑦)
28	27	adantrl 716	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑋 ⊆ 𝑦)
29		simprl 770	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → ∪ 𝑦 = ∪ 𝑋)
30	24, 12, 26, 28, 29	salgenss 46332	. . . . . . 7 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → (SalGen‘𝑋) ⊆ 𝑦)
31	22, 30	eqsstrd 3998	. . . . . 6 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 ⊆ 𝑦)
32	31	ex 412	. . . . 5 ⊢ (((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) → ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))
33	32	ralrimiva 3133	. . . 4 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))
34	20, 33	jca 511	. . 3 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)))
35	34	ex 412	. 2 ⊢ (𝜑 → ((SalGen‘𝑋) = 𝑆 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))))
36	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑋 ∈ 𝑉)
37		simprl1 1219	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑆 ∈ SAlg)
38		simprl2 1220	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → ∪ 𝑆 = ∪ 𝑋)
39		simprl3 1221	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑋 ⊆ 𝑆)
40		unieq 4899	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ∪ 𝑦 = ∪ 𝑤)
41	40	eqeq1d 2738	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (∪ 𝑦 = ∪ 𝑋 ↔ ∪ 𝑤 = ∪ 𝑋))
42		sseq2 3990	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑤))
43	41, 42	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) ↔ (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)))
44		sseq2 3990	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑆 ⊆ 𝑦 ↔ 𝑆 ⊆ 𝑤))
45	43, 44	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ↔ ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤)))
46	45	cbvralvw 3224	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ↔ ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
47	46	biimpi 216	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) → ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
48	47	adantr 480	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
49		simpr 484	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → 𝑤 ∈ SAlg)
50	48, 49	jca 511	. . . . . . . 8 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → (∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤) ∧ 𝑤 ∈ SAlg))
51	50	3ad2antr1 1189	. . . . . . 7 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → (∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤) ∧ 𝑤 ∈ SAlg))
52		3simpc 1150	. . . . . . . 8 ⊢ ((𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤))
53	52	adantl 481	. . . . . . 7 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤))
54		rspa 3235	. . . . . . 7 ⊢ ((∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤) ∧ 𝑤 ∈ SAlg) → ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
55	51, 53, 54	sylc 65	. . . . . 6 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → 𝑆 ⊆ 𝑤)
56	55	adantll 714	. . . . 5 ⊢ ((((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → 𝑆 ⊆ 𝑤)
57	56	adantll 714	. . . 4 ⊢ (((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → 𝑆 ⊆ 𝑤)
58	36, 37, 38, 39, 57	issalgend 46334	. . 3 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → (SalGen‘𝑋) = 𝑆)
59	58	ex 412	. 2 ⊢ (𝜑 → (((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)) → (SalGen‘𝑋) = 𝑆))
60	35, 59	impbid 212	1 ⊢ (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ⊆ wss 3931  ∪ cuni 4888  ‘cfv 6536  SAlgcsalg 46304  SalGencsalgen 46308

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-salg 46305  df-salgen 46309

This theorem is referenced by:  unisalgen2  46350