Theorem unidmex 42598

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 7750	. . 3 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
4		uniexg 7593	. . 3 ⊢ (dom 𝐹 ∈ V → ∪ dom 𝐹 ∈ V)
5	2, 3, 4	3syl 18	. 2 ⊢ (𝜑 → ∪ dom 𝐹 ∈ V)
6	1, 5	eqeltrid 2843	1 ⊢ (𝜑 → 𝑋 ∈ V)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∪ cuni 4839  dom cdm 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-cnv 5597  df-dm 5599  df-rn 5600

This theorem is referenced by:  omessle  44036  caragensplit  44038  omeunile  44043  caragenuncl  44051  omeunle  44054  omeiunlempt  44058  carageniuncllem2  44060  caragencmpl  44073