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 4127	. . . 4 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		dmrnssfld 5962	. . . 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 10724	1 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ∪ cun 3900   ⊆ wss 3902  ∪ cuni 4870  dom cdm 5659  ran crn 5660  WUnicwun 10712

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-cnv 5667  df-dm 5669  df-rn 5670  df-wun 10714

This theorem is used by:  wuncnv  10742  wunco  10745  wuntpos  10746  catcoppccl  18210