Theorem unidmex 45048

Description: If 𝐹 is a set, then ∪ dom 𝐹 is a set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
unidmex.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
unidmex.x	⊢ 𝑋 = ∪ dom 𝐹

Assertion

Ref	Expression
unidmex	⊢ (𝜑 → 𝑋 ∈ V)

Proof of Theorem unidmex

Step	Hyp	Ref	Expression
1		unidmex.x	. 2 ⊢ 𝑋 = ∪ dom 𝐹
2		unidmex.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
3		dmexg 7834	. . 3 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
4		uniexg 7676	. . 3 ⊢ (dom 𝐹 ∈ V → ∪ dom 𝐹 ∈ V)
5	2, 3, 4	3syl 18	. 2 ⊢ (𝜑 → ∪ dom 𝐹 ∈ V)
6	1, 5	eqeltrid 2832	1 ⊢ (𝜑 → 𝑋 ∈ V)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∪ cuni 4858  dom cdm 5619

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-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-cnv 5627  df-dm 5629  df-rn 5630

This theorem is referenced by:  omessle  46499  caragensplit  46501  omeunile  46506  caragenuncl  46514  omeunle  46517  omeiunlempt  46521  carageniuncllem2  46523  caragencmpl  46536