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Theorem wunrn 10706
Description: A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wunrn (𝜑 → ran 𝐴𝑈)

Proof of Theorem wunrn
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunop.2 . . . 4 (𝜑𝐴𝑈)
31, 2wununi 10683 . . 3 (𝜑 𝐴𝑈)
41, 3wununi 10683 . 2 (𝜑 𝐴𝑈)
5 ssun2 4169 . . . 4 ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 dmrnssfld 5961 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
75, 6sstri 3987 . . 3 ran 𝐴 𝐴
87a1i 11 . 2 (𝜑 → ran 𝐴 𝐴)
91, 4, 8wunss 10689 1 (𝜑 → ran 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cun 3942  wss 3944   cuni 4901  dom cdm 5669  ran crn 5670  WUnicwun 10677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-tr 5259  df-cnv 5677  df-dm 5679  df-rn 5680  df-wun 10679
This theorem is referenced by:  wuncnv  10707  wunfv  10709  wunco  10710  wuntpos  10711  wunstr  17103  wunfunc  17831  wunfuncOLD  17832  wunnat  17889  wunnatOLD  17890  catcoppccl  18049  catcoppcclOLD  18050  catcfuccl  18051  catcfucclOLD  18052  catcxpccl  18141  catcxpcclOLD  18142
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