Theorem dfsalgen2 46784

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 2745	. . . . . . 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 46775	. . . . . . . 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 2839	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 ∈ SAlg)
9		unieq 4849	. . . . . . 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 2739	. . . . . . 7 ⊢ (SalGen‘𝑋) = (SalGen‘𝑋)
13		eqid 2739	. . . . . . 7 ⊢ ∪ 𝑋 = ∪ 𝑋
14	11, 12, 13	salgenuni 46780	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ (SalGen‘𝑋) = ∪ 𝑋)
15	10, 14	eqtr3d 2776	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∪ 𝑆 = ∪ 𝑋)
16	12	sssalgen 46778	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (SalGen‘𝑋))
17	11, 16	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ (SalGen‘𝑋))
18		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
19	17, 18	sseqtrd 3951	. . . . 5 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ 𝑆)
20	8, 15, 19	3jca 1134	. . . 4 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))
21	3	ad2antrr 732	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑆 = (SalGen‘𝑋))
22	21	adantrl 722	. . . . . . 7 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 = (SalGen‘𝑋))
23	11	ad2antrr 732	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ∈ 𝑉)
24	23	adantrl 722	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑋 ∈ 𝑉)
25		simplr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ SAlg)
26	25	adantrl 722	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑦 ∈ SAlg)
27		simpr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ⊆ 𝑦)
28	27	adantrl 722	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑋 ⊆ 𝑦)
29		simprl 776	. . . . . . . 8 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → ∪ 𝑦 = ∪ 𝑋)
30	24, 12, 26, 28, 29	salgenss 46779	. . . . . . 7 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → (SalGen‘𝑋) ⊆ 𝑦)
31	22, 30	eqsstrd 3949	. . . . . 6 ⊢ ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 ⊆ 𝑦)
32	31	ex 413	. . . . 5 ⊢ (((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) → ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))
33	32	ralrimiva 3131	. . . 4 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))
34	20, 33	jca 516	. . 3 ⊢ ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)))
35	34	ex 413	. 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 1225	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑆 ∈ SAlg)
38		simprl2 1226	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → ∪ 𝑆 = ∪ 𝑋)
39		simprl3 1227	. . . 4 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → 𝑋 ⊆ 𝑆)
40		unieq 4849	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → ∪ 𝑦 = ∪ 𝑤)
41	40	eqeq1d 2741	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (∪ 𝑦 = ∪ 𝑋 ↔ ∪ 𝑤 = ∪ 𝑋))
42		sseq2 3941	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑤))
43	41, 42	anbi12d 638	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) ↔ (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)))
44		sseq2 3941	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑆 ⊆ 𝑦 ↔ 𝑆 ⊆ 𝑤))
45	43, 44	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ↔ ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤)))
46	45	cbvralvw 3217	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ↔ ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
47	46	birani 504	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → ∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
48		simpr 485	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → 𝑤 ∈ SAlg)
49	47, 48	jca 516	. . . . . . . 8 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ 𝑤 ∈ SAlg) → (∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤) ∧ 𝑤 ∈ SAlg))
50	49	3ad2antr1 1195	. . . . . . 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))
51		3simpc 1156	. . . . . . . 8 ⊢ ((𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤))
52	51	adantl 482	. . . . . . 7 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → (∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤))
53		rspa 3228	. . . . . . 7 ⊢ ((∀𝑤 ∈ SAlg ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤) ∧ 𝑤 ∈ SAlg) → ((∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤) → 𝑆 ⊆ 𝑤))
54	50, 52, 53	sylc 65	. . . . . 6 ⊢ ((∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → 𝑆 ⊆ 𝑤)
55	54	adantll 720	. . . . 5 ⊢ ((((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → 𝑆 ⊆ 𝑤)
56	55	adantll 720	. . . 4 ⊢ (((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) ∧ (𝑤 ∈ SAlg ∧ ∪ 𝑤 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑤)) → 𝑆 ⊆ 𝑤)
57	36, 37, 38, 39, 56	issalgend 46781	. . 3 ⊢ ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))) → (SalGen‘𝑋) = 𝑆)
58	57	ex 413	. 2 ⊢ (𝜑 → (((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)) → (SalGen‘𝑋) = 𝑆))
59	35, 58	impbid 213	1 ⊢ (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ⊆ wss 3883  ∪ cuni 4838  ‘cfv 6485  SAlgcsalg 46751  SalGencsalgen 46755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-salg 46752  df-salgen 46756

This theorem is referenced by:  unisalgen2  46797