Theorem sssalgen 45956

Description: A set is a subset of the sigma-algebra it generates. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypothesis

Ref	Expression
sssalgen.1	⊢ 𝑆 = (SalGen‘𝑋)

Assertion

Ref	Expression
sssalgen	⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝑆)

Proof of Theorem sssalgen

Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 4972	. . . 4 ⊢ (𝑋 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ ∀𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}𝑋 ⊆ 𝑡)
2		unieq 4924	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → ∪ 𝑠 = ∪ 𝑡)
3	2	eqeq1d 2728	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑡 = ∪ 𝑋))
4		sseq2 4006	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑡))
5	3, 4	anbi12d 630	. . . . . . 7 ⊢ (𝑠 = 𝑡 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
6	5	elrab 3681	. . . . . 6 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
7	6	biimpi 215	. . . . 5 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
8	7	simprrd 772	. . . 4 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → 𝑋 ⊆ 𝑡)
9	1, 8	mprgbir 3058	. . 3 ⊢ 𝑋 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}
10	9	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
11		sssalgen.1	. . 3 ⊢ 𝑆 = (SalGen‘𝑋)
12		salgenval 45942	. . 3 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
13	11, 12	eqtr2id 2779	. 2 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} = 𝑆)
14	10, 13	sseqtrd 4020	1 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝑆)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  {crab 3419   ⊆ wss 3947  ∪ cuni 4913  ∩ cint 4954  ‘cfv 6554  SAlgcsalg 45929  SalGencsalgen 45933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562  df-salg 45930  df-salgen 45934

This theorem is referenced by:  dfsalgen2  45962  iooborel  45972  opnssborel  46256  cnfsmf  46361