Theorem issalgend 47455

Description: One side of dfsalgen2 47458. 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 47453	. 2 ⊢ (𝜑 → (SalGen‘𝑋) ⊆ 𝑆)
7		simpl 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → 𝜑)
8		elrabi 3640	. . . . . . 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 4877	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑦 → ∪ 𝑠 = ∪ 𝑦)
11	10	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑦 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑦 = ∪ 𝑋))
12		sseq2 3935	. . . . . . . . . . 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 3644	. . . . . . . . 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 4923	. . . 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 47438	. . . 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 3946	. 2 ⊢ (𝜑 → 𝑆 ⊆ (SalGen‘𝑋))
28	6, 27	eqssd 3926	1 ⊢ (𝜑 → (SalGen‘𝑋) = 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3896  ∪ cuni 4866  ∩ cint 4906  ‘cfv 6541  SAlgcsalg 47425  SalGencsalgen 47429

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6496  df-fun 6543  df-fv 6549  df-salg 47426  df-salgen 47430

This theorem is used by:  dfsalgen2  47458