Theorem unisalgen 46296

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 46293	. . 3 ⊢ (𝜑 → ∪ 𝑆 = 𝑈)
5	4	eqcomd 2741	. 2 ⊢ (𝜑 → 𝑈 = ∪ 𝑆)
6	2	a1i 11	. . . 4 ⊢ (𝜑 → 𝑆 = (SalGen‘𝑋))
7		salgencl 46288	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg)
8	1, 7	syl 17	. . . 4 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg)
9	6, 8	eqeltrd 2839	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
10		saluni 46281	. . 3 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆)
11	9, 10	syl 17	. 2 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
12	5, 11	eqeltrd 2839	1 ⊢ (𝜑 → 𝑈 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∪ cuni 4912  ‘cfv 6563  SAlgcsalg 46264  SalGencsalgen 46268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-salg 46265  df-salgen 46269

This theorem is referenced by:  salgensscntex  46300