Theorem unisalgen 46345

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 46342	. . 3 ⊢ (𝜑 → ∪ 𝑆 = 𝑈)
5	4	eqcomd 2736	. 2 ⊢ (𝜑 → 𝑈 = ∪ 𝑆)
6	2	a1i 11	. . . 4 ⊢ (𝜑 → 𝑆 = (SalGen‘𝑋))
7		salgencl 46337	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg)
8	1, 7	syl 17	. . . 4 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg)
9	6, 8	eqeltrd 2829	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
10		saluni 46330	. . 3 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆)
11	9, 10	syl 17	. 2 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
12	5, 11	eqeltrd 2829	1 ⊢ (𝜑 → 𝑈 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∪ cuni 4874  ‘cfv 6514  SAlgcsalg 46313  SalGencsalgen 46317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-salg 46314  df-salgen 46318

This theorem is referenced by:  salgensscntex  46349