Theorem issalgend 44569

Description: One side of dfsalgen2 44572. 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 2736	. . 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 44567	. 2 ⊢ (𝜑 → (SalGen‘𝑋) ⊆ 𝑆)
7		simpl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝜑)
8		elrabi 3639	. . . . . . 7 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → 𝑦 ∈ SAlg)
9	8	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝑦 ∈ SAlg)
10		unieq 4876	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑦 → ∪ 𝑠 = ∪ 𝑦)
11	10	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑦 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑦 = ∪ 𝑋))
12		sseq2 3970	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑦 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑦))
13	11, 12	anbi12d 631	. . . . . . . . . 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 215	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → (𝑦 ∈ SAlg ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)))
16	15	simprld 770	. . . . . . 7 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑦 = ∪ 𝑋)
17	16	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → ∪ 𝑦 = ∪ 𝑋)
18	15	simprrd 772	. . . . . . 7 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → 𝑋 ⊆ 𝑦)
19	18	adantl 482	. . . . . 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 1372	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝑆 ⊆ 𝑦)
22	21	ralrimiva 3143	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}𝑆 ⊆ 𝑦)
23		ssint 4925	. . . 4 ⊢ (𝑆 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}𝑆 ⊆ 𝑦)
24	22, 23	sylibr 233	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
25		salgenval 44552	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
26	1, 25	syl 17	. . 3 ⊢ (𝜑 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
27	24, 26	sseqtrrd 3985	. 2 ⊢ (𝜑 → 𝑆 ⊆ (SalGen‘𝑋))
28	6, 27	eqssd 3961	1 ⊢ (𝜑 → (SalGen‘𝑋) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  {crab 3407   ⊆ wss 3910  ∪ cuni 4865  ∩ cint 4907  ‘cfv 6496  SAlgcsalg 44539  SalGencsalgen 44543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-salg 44540  df-salgen 44544

This theorem is referenced by:  dfsalgen2  44572