Theorem issalgend 46788

Description: One side of dfsalgen2 46791. 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 2740	. . 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 46786	. 2 ⊢ (𝜑 → (SalGen‘𝑋) ⊆ 𝑆)
7		simpl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝜑)
8		elrabi 3632	. . . . . . 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 4856	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑦 → ∪ 𝑠 = ∪ 𝑦)
11	10	eqeq1d 2742	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑦 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑦 = ∪ 𝑋))
12		sseq2 3948	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑦 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑦))
13	11, 12	anbi12d 638	. . . . . . . . . 10 ⊢ (𝑠 = 𝑦 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)))
14	13	elrab 3636	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑦 ∈ SAlg ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)))
15	14	biimpi 217	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → (𝑦 ∈ SAlg ∧ (∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)))
16	15	simprld 777	. . . . . . 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 779	. . . . . . 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 1380	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝑆 ⊆ 𝑦)
22	21	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}𝑆 ⊆ 𝑦)
23		ssint 4901	. . . 4 ⊢ (𝑆 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}𝑆 ⊆ 𝑦)
24	22, 23	sylibr 235	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
25		salgenval 46771	. . . 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 3959	. 2 ⊢ (𝜑 → 𝑆 ⊆ (SalGen‘𝑋))
28	6, 27	eqssd 3939	1 ⊢ (𝜑 → (SalGen‘𝑋) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {crab 3392   ⊆ wss 3890  ∪ cuni 4845  ∩ cint 4884  ‘cfv 6492  SAlgcsalg 46758  SalGencsalgen 46762

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 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

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 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-salg 46759  df-salgen 46763

This theorem is referenced by:  dfsalgen2  46791