Theorem wundm 10753

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 10731	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
4	1, 3	wununi 10731	. 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈)
5		ssun1 4170	. . . 4 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		dmrnssfld 5973	. . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴
7	5, 6	sstri 3986	. . 3 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴
8	7	a1i 11	. 2 ⊢ (𝜑 → dom 𝐴 ⊆ ∪ ∪ 𝐴)
9	1, 4, 8	wunss 10737	1 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2098   ∪ cun 3942   ⊆ wss 3944  ∪ cuni 4909  dom cdm 5678  ran crn 5679  WUnicwun 10725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-cnv 5686  df-dm 5688  df-rn 5689  df-wun 10727

This theorem is referenced by:  wuncnv  10755  wunco  10758  wuntpos  10759  catcoppccl  18109  catcoppcclOLD  18110