Theorem dfsalgen2 40559

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 2628	. . . . . . 7 ⊢ ((SalGen‘𝑋) = 𝑆 → 𝑆 = (SalGen‘𝑋))
3	2	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = (SalGen‘𝑋))
4		dfsalgen2.1	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
5		salgencl 40550	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg)
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg)
7	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) ∈ SAlg)
8	3, 7	eqeltrd 2701	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 ∈ SAlg)
9		unieq 4444	. . . . . . 7 ⊢ ((SalGen‘𝑋) = 𝑆 → ∪ (SalGen‘𝑋) = ∪ 𝑆)
10	9	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ (SalGen‘𝑋) = ∪ 𝑆)
11	4	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ∈ 𝑉)
12		eqid 2622	. . . . . . 7 ⊢ (SalGen‘𝑋) = (SalGen‘𝑋)
13		eqid 2622	. . . . . . 7 ⊢ ∪ 𝑋 = ∪ 𝑋
14	11, 12, 13	salgenuni 40555	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ (SalGen‘𝑋) = ∪ 𝑋)
15	10, 14	eqtr3d 2658	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ 𝑆 = ∪ 𝑋)
16	12	sssalgen 40553	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (SalGen‘𝑋))
17	11, 16	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ (SalGen‘𝑋))
18		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
19	17, 18	sseqtrd 3641	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ 𝑆)
20	8, 15, 19	3jca 1242	. . . 4 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))
21	3	ad2antrr 762	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑆 = (SalGen‘𝑋))
22	21	adantrl 752	. . . . . . 7 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 = (SalGen‘𝑋))
23	11	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ∈ 𝑉)
24	23	adantrl 752	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑋 ∈ 𝑉)
25		simplr 792	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ SAlg)
26	25	adantrl 752	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑦 ∈ SAlg)
27		simpr 477	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ⊆ 𝑦)
28	27	adantrl 752	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑋 ⊆ 𝑦)
29		simprl 794	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → ∪ 𝑦 = ∪ 𝑋)
30	24, 12, 26, 28, 29	salgenss 40554	. . . . . . 7 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → (SalGen‘𝑋) ⊆ 𝑦)
31	22, 30	eqsstrd 3639	. . . . . 6 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 ⊆ 𝑦)
32	31	ex 450	. . . . 5 ⊢ (((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) → ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))
33	32	ralrimiva 2966	. . . 4 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))
34	20, 33	jca 554	. . 3 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)))
35	34	ex 450	. 2 ⊢ (𝜑 → ((SalGen‘𝑋) = 𝑆 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))))
36	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑋 ∈ 𝑉)
37		simprl1 1106	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑆 ∈ SAlg)
38		simprl2 1107	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → ∪ 𝑆 = ∪ 𝑋)
39		simprl3 1108	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑋 ⊆ 𝑆)
40		unieq 4444	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ∪ 𝑦 = ∪ 𝑤)
41	40	eqeq1d 2624	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (∪ 𝑦 = ∪ 𝑋 ↔ ∪ 𝑤 = ∪ 𝑋))
42		sseq2 3627	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑤))
43	41, 42	anbi12d 747	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) ↔ (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)))
44		sseq2 3627	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑆 ⊆ 𝑦 ↔ 𝑆 ⊆ 𝑤))
45	43, 44	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ↔ ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤)))
46	45	cbvralv 3171	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ↔ ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
47	46	biimpi 206	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) → ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
48	47	adantr 481	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
49		simpr 477	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → 𝑤 ∈ SAlg)
50	48, 49	jca 554	. . . . . . . 8 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → (∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤) ∧ 𝑤 ∈ SAlg))
51	50	3ad2antr1 1226	. . . . . . 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 1060	. . . . . . . 8 ⊢ ((𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤))
53	52	adantl 482	. . . . . . 7 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤))
54		rspa 2930	. . . . . . 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 750	. . . . 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 750	. . . 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 40556	. . 3 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → (SalGen‘𝑋) = 𝑆)
59	58	ex 450	. 2 ⊢ (𝜑 → (((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)) → (SalGen‘𝑋) = 𝑆))
60	35, 59	impbid 202	1 ⊢ (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  ∪ cuni 4436  ‘cfv 5888  SAlgcsalg 40528  SalGencsalgen 40532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-salg 40529  df-salgen 40533

This theorem is referenced by:  unisalgen2  40572