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Theorem wunrn 9430
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 9407 . . 3 (𝜑 𝐴𝑈)
41, 3wununi 9407 . 2 (𝜑 𝐴𝑈)
5 ssun2 3739 . . . 4 ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 dmrnssfld 5305 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
75, 6sstri 3577 . . 3 ran 𝐴 𝐴
87a1i 11 . 2 (𝜑 → ran 𝐴 𝐴)
91, 4, 8wunss 9413 1 (𝜑 → ran 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  cun 3538  wss 3540   cuni 4372  dom cdm 5038  ran crn 5039  WUnicwun 9401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-cnv 5046  df-dm 5048  df-rn 5049  df-wun 9403
This theorem is referenced by:  wuncnv  9431  wunfv  9433  wunco  9434  wuntpos  9435  wunstr  15714  wunfunc  16382  wunnat  16439  catcoppccl  16581  catcfuccl  16582  catcxpccl  16670
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