Theorem issalgend 47035

Description: One side of dfsalgen2 47038. 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 2763	. . 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 47033	. 2 ⊢ (𝜑 → (SalGen‘𝑋) ⊆ 𝑆)
7		simpl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝜑)
8		elrabi 3647	. . . . . . 7 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → 𝑦 ∈ SAlg)
9	8	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝑦 ∈ SAlg)
10		unieq 4884	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑦 → ∪ 𝑠 = ∪ 𝑦)
11	10	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑦 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑦 = ∪ 𝑋))
12		sseq2 3964	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑦 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑦))
13	11, 12	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑠 = 𝑦 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)))
14	13	elrab 3651	. . . . . . . . 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 783	. . . . . . 7 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑦 = ∪ 𝑋)
17	16	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → ∪ 𝑦 = ∪ 𝑋)
18	15	simprrd 785	. . . . . . 7 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → 𝑋 ⊆ 𝑦)
19	18	adantl 486	. . . . . 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 3157	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}𝑆 ⊆ 𝑦)
23		ssint 4930	. . . 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 47018	. . . 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 3975	. 2 ⊢ (𝜑 → 𝑆 ⊆ (SalGen‘𝑋))
28	6, 27	eqssd 3955	1 ⊢ (𝜑 → (SalGen‘𝑋) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416   ⊆ wss 3906  ∪ cuni 4873  ∩ cint 4913  ‘cfv 6538  SAlgcsalg 47005  SalGencsalgen 47009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-salg 47006  df-salgen 47010

This theorem is referenced by:  dfsalgen2  47038