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Theorem wunrn 10724
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 10701 . . 3 (𝜑 𝐴𝑈)
41, 3wununi 10701 . 2 (𝜑 𝐴𝑈)
5 ssun2 4174 . . . 4 ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 dmrnssfld 5970 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
75, 6sstri 3992 . . 3 ran 𝐴 𝐴
87a1i 11 . 2 (𝜑 → ran 𝐴 𝐴)
91, 4, 8wunss 10707 1 (𝜑 → ran 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cun 3947  wss 3949   cuni 4909  dom cdm 5677  ran crn 5678  WUnicwun 10695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-cnv 5685  df-dm 5687  df-rn 5688  df-wun 10697
This theorem is referenced by:  wuncnv  10725  wunfv  10727  wunco  10728  wuntpos  10729  wunstr  17121  wunfunc  17849  wunfuncOLD  17850  wunnat  17907  wunnatOLD  17908  catcoppccl  18067  catcoppcclOLD  18068  catcfuccl  18069  catcfucclOLD  18070  catcxpccl  18159  catcxpcclOLD  18160
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