Theorem issalgend 47166

Description: One side of dfsalgen2 47169. If a sigma-algebra on ∪ 𝑋 includes 𝑋 and it is included in all the sigma-algebras with such two properties, then it is the sigma-algebra generated by 𝑋. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
issalgend.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
issalgend.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
issalgend.u	⊢ (𝜑 → ∪ 𝑆 = ∪ 𝑋)
issalgend.i	⊢ (𝜑 → 𝑋 ⊆ 𝑆)
issalgend.a	⊢ ((𝜑 ∧ (𝑦 ∈ SAlg ∧ ∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 ⊆ 𝑦)

Assertion

Ref	Expression
issalgend	⊢ (𝜑 → (SalGen‘𝑋) = 𝑆)

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

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem issalgend

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issalgend.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		eqid 2760	. . 3 ⊢ (SalGen‘𝑋) = (SalGen‘𝑋)
3		issalgend.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
4		issalgend.i	. . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
5		issalgend.u	. . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝑋)
6	1, 2, 3, 4, 5	salgenss 47164	. 2 ⊢ (𝜑 → (SalGen‘𝑋) ⊆ 𝑆)
7		simpl 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝜑)
8		elrabi 3641	. . . . . . 7 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → 𝑦 ∈ SAlg)
9	8	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝑦 ∈ SAlg)
10		unieq 4878	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑦 → ∪ 𝑠 = ∪ 𝑦)
11	10	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑦 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑦 = ∪ 𝑋))
12		sseq2 3957	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑦 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑦))
13	11, 12	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑠 = 𝑦 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)))
14	13	elrab 3645	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑦 ∈ SAlg ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)))
15	14	biimpi 219	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → (𝑦 ∈ SAlg ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)))
16	15	simprld 784	. . . . . . 7 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑦 = ∪ 𝑋)
17	16	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → ∪ 𝑦 = ∪ 𝑋)
18	15	simprrd 786	. . . . . . 7 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → 𝑋 ⊆ 𝑦)
19	18	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝑋 ⊆ 𝑦)
20		issalgend.a	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ SAlg ∧ ∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 ⊆ 𝑦)
21	7, 9, 17, 19, 20	syl13anc 1399	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝑆 ⊆ 𝑦)
22	21	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}𝑆 ⊆ 𝑦)
23		ssint 4924	. . . 4 ⊢ (𝑆 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}𝑆 ⊆ 𝑦)
24	22, 23	sylibr 237	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
25		salgenval 47149	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
26	1, 25	syl 18	. . 3 ⊢ (𝜑 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
27	24, 26	sseqtrrd 3968	. 2 ⊢ (𝜑 → 𝑆 ⊆ (SalGen‘𝑋))
28	6, 27	eqssd 3948	1 ⊢ (𝜑 → (SalGen‘𝑋) = 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412   ⊆ wss 3899  ∪ cuni 4867  ∩ cint 4907  ‘cfv 6533  SAlgcsalg 47136  SalGencsalgen 47140

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-salg 47137  df-salgen 47141

This theorem is used by:  dfsalgen2  47169