Theorem wundm 10765

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 10743	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
4	1, 3	wununi 10743	. 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈)
5		ssun1 4187	. . . 4 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		dmrnssfld 5986	. . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴
7	5, 6	sstri 4004	. . 3 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴
8	7	a1i 11	. 2 ⊢ (𝜑 → dom 𝐴 ⊆ ∪ ∪ 𝐴)
9	1, 4, 8	wunss 10749	1 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2105   ∪ cun 3960   ⊆ wss 3962  ∪ cuni 4911  dom cdm 5688  ran crn 5689  WUnicwun 10737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-cnv 5696  df-dm 5698  df-rn 5699  df-wun 10739

This theorem is referenced by:  wuncnv  10767  wunco  10770  wuntpos  10771  catcoppccl  18170  catcoppcclOLD  18171