Theorem unisalgen 43554

Description: The union of a set belongs to the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
unisalgen.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
unisalgen.s	⊢ 𝑆 = (SalGen‘𝑋)
unisalgen.u	⊢ 𝑈 = ∪ 𝑋

Assertion

Ref	Expression
unisalgen	⊢ (𝜑 → 𝑈 ∈ 𝑆)

Proof of Theorem unisalgen

Step	Hyp	Ref	Expression
1		unisalgen.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		unisalgen.s	. . . 4 ⊢ 𝑆 = (SalGen‘𝑋)
3		unisalgen.u	. . . 4 ⊢ 𝑈 = ∪ 𝑋
4	1, 2, 3	salgenuni 43551	. . 3 ⊢ (𝜑 → ∪ 𝑆 = 𝑈)
5	4	eqcomd 2743	. 2 ⊢ (𝜑 → 𝑈 = ∪ 𝑆)
6	2	a1i 11	. . . 4 ⊢ (𝜑 → 𝑆 = (SalGen‘𝑋))
7		salgencl 43546	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg)
8	1, 7	syl 17	. . . 4 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg)
9	6, 8	eqeltrd 2838	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
10		saluni 43540	. . 3 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆)
11	9, 10	syl 17	. 2 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
12	5, 11	eqeltrd 2838	1 ⊢ (𝜑 → 𝑈 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  ∪ cuni 4819  ‘cfv 6380  SAlgcsalg 43524  SalGencsalgen 43528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-salg 43525  df-salgen 43529

This theorem is referenced by:  salgensscntex  43558