Theorem wundm 10639

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 10617	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
4	1, 3	wununi 10617	. 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈)
5		ssun1 4130	. . . 4 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		dmrnssfld 5923	. . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴
7	5, 6	sstri 3943	. . 3 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴
8	7	a1i 11	. 2 ⊢ (𝜑 → dom 𝐴 ⊆ ∪ ∪ 𝐴)
9	1, 4, 8	wunss 10623	1 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2113   ∪ cun 3899   ⊆ wss 3901  ∪ cuni 4863  dom cdm 5624  ran crn 5625  WUnicwun 10611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-cnv 5632  df-dm 5634  df-rn 5635  df-wun 10613

This theorem is referenced by:  wuncnv  10641  wunco  10644  wuntpos  10645  catcoppccl  18041