Theorem unidmex 45074

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 7897	. . 3 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
4		uniexg 7734	. . 3 ⊢ (dom 𝐹 ∈ V → ∪ dom 𝐹 ∈ V)
5	2, 3, 4	3syl 18	. 2 ⊢ (𝜑 → ∪ dom 𝐹 ∈ V)
6	1, 5	eqeltrid 2838	1 ⊢ (𝜑 → 𝑋 ∈ V)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ∪ cuni 4883  dom cdm 5654

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-cnv 5662  df-dm 5664  df-rn 5665

This theorem is referenced by:  omessle  46527  caragensplit  46529  omeunile  46534  caragenuncl  46542  omeunle  46545  omeiunlempt  46549  carageniuncllem2  46551  caragencmpl  46564