Theorem dfsalgen2 42644

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 2827	. . . . . . 7 ⊢ ((SalGen‘𝑋) = 𝑆 → 𝑆 = (SalGen‘𝑋))
3	2	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = (SalGen‘𝑋))
4		dfsalgen2.1	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
5		salgencl 42635	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg)
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg)
7	6	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) ∈ SAlg)
8	3, 7	eqeltrd 2913	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 ∈ SAlg)
9		unieq 4849	. . . . . . 7 ⊢ ((SalGen‘𝑋) = 𝑆 → ∪ (SalGen‘𝑋) = ∪ 𝑆)
10	9	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ (SalGen‘𝑋) = ∪ 𝑆)
11	4	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ∈ 𝑉)
12		eqid 2821	. . . . . . 7 ⊢ (SalGen‘𝑋) = (SalGen‘𝑋)
13		eqid 2821	. . . . . . 7 ⊢ ∪ 𝑋 = ∪ 𝑋
14	11, 12, 13	salgenuni 42640	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ (SalGen‘𝑋) = ∪ 𝑋)
15	10, 14	eqtr3d 2858	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ 𝑆 = ∪ 𝑋)
16	12	sssalgen 42638	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (SalGen‘𝑋))
17	11, 16	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ (SalGen‘𝑋))
18		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
19	17, 18	sseqtrd 4007	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ 𝑆)
20	8, 15, 19	3jca 1124	. . . 4 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))
21	3	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑆 = (SalGen‘𝑋))
22	21	adantrl 714	. . . . . . 7 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 = (SalGen‘𝑋))
23	11	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ∈ 𝑉)
24	23	adantrl 714	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑋 ∈ 𝑉)
25		simplr 767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ SAlg)
26	25	adantrl 714	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑦 ∈ SAlg)
27		simpr 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ⊆ 𝑦)
28	27	adantrl 714	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑋 ⊆ 𝑦)
29		simprl 769	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → ∪ 𝑦 = ∪ 𝑋)
30	24, 12, 26, 28, 29	salgenss 42639	. . . . . . 7 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → (SalGen‘𝑋) ⊆ 𝑦)
31	22, 30	eqsstrd 4005	. . . . . 6 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 ⊆ 𝑦)
32	31	ex 415	. . . . 5 ⊢ (((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) → ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))
33	32	ralrimiva 3182	. . . 4 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))
34	20, 33	jca 514	. . 3 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)))
35	34	ex 415	. 2 ⊢ (𝜑 → ((SalGen‘𝑋) = 𝑆 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))))
36	4	adantr 483	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑋 ∈ 𝑉)
37		simprl1 1214	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑆 ∈ SAlg)
38		simprl2 1215	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → ∪ 𝑆 = ∪ 𝑋)
39		simprl3 1216	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑋 ⊆ 𝑆)
40		unieq 4849	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ∪ 𝑦 = ∪ 𝑤)
41	40	eqeq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (∪ 𝑦 = ∪ 𝑋 ↔ ∪ 𝑤 = ∪ 𝑋))
42		sseq2 3993	. . . . . . . . . . . . . 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 3993	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑆 ⊆ 𝑦 ↔ 𝑆 ⊆ 𝑤))
45	43, 44	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ↔ ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤)))
46	45	cbvralvw 3449	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ↔ ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
47	46	biimpi 218	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) → ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
48	47	adantr 483	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
49		simpr 487	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → 𝑤 ∈ SAlg)
50	48, 49	jca 514	. . . . . . . 8 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → (∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤) ∧ 𝑤 ∈ SAlg))
51	50	3ad2antr1 1184	. . . . . . 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 1146	. . . . . . . 8 ⊢ ((𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤))
53	52	adantl 484	. . . . . . 7 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤))
54		rspa 3206	. . . . . . 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 712	. . . . 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 712	. . . 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 42641	. . 3 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → (SalGen‘𝑋) = 𝑆)
59	58	ex 415	. 2 ⊢ (𝜑 → (((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)) → (SalGen‘𝑋) = 𝑆))
60	35, 59	impbid 214	1 ⊢ (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ⊆ wss 3936  ∪ cuni 4838  ‘cfv 6355  SAlgcsalg 42613  SalGencsalgen 42617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-int 4877  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-salg 42614  df-salgen 42618

This theorem is referenced by:  unisalgen2  42657