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Theorem wunrn 10140
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 10117 . . 3 (𝜑 𝐴𝑈)
41, 3wununi 10117 . 2 (𝜑 𝐴𝑈)
5 ssun2 4148 . . . 4 ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 dmrnssfld 5835 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
75, 6sstri 3975 . . 3 ran 𝐴 𝐴
87a1i 11 . 2 (𝜑 → ran 𝐴 𝐴)
91, 4, 8wunss 10123 1 (𝜑 → ran 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cun 3933  wss 3935   cuni 4832  dom cdm 5549  ran crn 5550  WUnicwun 10111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-tr 5165  df-cnv 5557  df-dm 5559  df-rn 5560  df-wun 10113
This theorem is referenced by:  wuncnv  10141  wunfv  10143  wunco  10144  wuntpos  10145  wunstr  16497  wunfunc  17159  wunnat  17216  catcoppccl  17358  catcfuccl  17359  catcxpccl  17447
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