Theorem wundm 10740

Description: A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wun0.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wunop.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
wundm	⊢ (𝜑 → dom 𝐴 ∈ 𝑈)

Proof of Theorem wundm

Step	Hyp	Ref	Expression
1		wun0.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wunop.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
3	1, 2	wununi 10718	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
4	1, 3	wununi 10718	. 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈)
5		ssun1 4153	. . . 4 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		dmrnssfld 5953	. . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴
7	5, 6	sstri 3968	. . 3 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴
8	7	a1i 11	. 2 ⊢ (𝜑 → dom 𝐴 ⊆ ∪ ∪ 𝐴)
9	1, 4, 8	wunss 10724	1 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2108   ∪ cun 3924   ⊆ wss 3926  ∪ cuni 4883  dom cdm 5654  ran crn 5655  WUnicwun 10712

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

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-ne 2933  df-ral 3052  df-rex 3061  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-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-tr 5230  df-cnv 5662  df-dm 5664  df-rn 5665  df-wun 10714

This theorem is referenced by:  wuncnv  10742  wunco  10745  wuntpos  10746  catcoppccl  18128