Theorem unidmex 41305

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 7607	. . 3 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
4		uniexg 7460	. . 3 ⊢ (dom 𝐹 ∈ V → ∪ dom 𝐹 ∈ V)
5	2, 3, 4	3syl 18	. 2 ⊢ (𝜑 → ∪ dom 𝐹 ∈ V)
6	1, 5	eqeltrid 2917	1 ⊢ (𝜑 → 𝑋 ∈ V)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  Vcvv 3495  ∪ cuni 4832  dom cdm 5550

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-cnv 5558  df-dm 5560  df-rn 5561

This theorem is referenced by:  omessle  42773  caragensplit  42775  omeunile  42780  caragenuncl  42788  omeunle  42791  omeiunlempt  42795  carageniuncllem2  42797  caragencmpl  42810