Theorem unidmex 45439

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 7855	. . 3 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
4		uniexg 7697	. . 3 ⊢ (dom 𝐹 ∈ V → ∪ dom 𝐹 ∈ V)
5	2, 3, 4	3syl 18	. 2 ⊢ (𝜑 → ∪ dom 𝐹 ∈ V)
6	1, 5	eqeltrid 2841	1 ⊢ (𝜑 → 𝑋 ∈ V)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∪ cuni 4865  dom cdm 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-cnv 5642  df-dm 5644  df-rn 5645

This theorem is referenced by:  omessle  46885  caragensplit  46887  omeunile  46892  caragenuncl  46900  omeunle  46903  omeiunlempt  46907  carageniuncllem2  46909  caragencmpl  46922