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Theorem wundm 9865
 Description: A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wundm (𝜑 → dom 𝐴𝑈)

Proof of Theorem wundm
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunop.2 . . . 4 (𝜑𝐴𝑈)
31, 2wununi 9843 . . 3 (𝜑 𝐴𝑈)
41, 3wununi 9843 . 2 (𝜑 𝐴𝑈)
5 ssun1 4003 . . . 4 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 dmrnssfld 5617 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
75, 6sstri 3836 . . 3 dom 𝐴 𝐴
87a1i 11 . 2 (𝜑 → dom 𝐴 𝐴)
91, 4, 8wunss 9849 1 (𝜑 → dom 𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2164   ∪ cun 3796   ⊆ wss 3798  ∪ cuni 4658  dom cdm 5342  ran crn 5343  WUnicwun 9837 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-tr 4976  df-cnv 5350  df-dm 5352  df-rn 5353  df-wun 9839 This theorem is referenced by:  wuncnv  9867  wunco  9870  wuntpos  9871  catcoppccl  17110
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