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Theorem wunrn 9096
Description: A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1  |-  ( ph  ->  U  e. WUni )
wunop.2  |-  ( ph  ->  A  e.  U )
Assertion
Ref Expression
wunrn  |-  ( ph  ->  ran  A  e.  U
)

Proof of Theorem wunrn
StepHypRef Expression
1 wun0.1 . 2  |-  ( ph  ->  U  e. WUni )
2 wunop.2 . . . 4  |-  ( ph  ->  A  e.  U )
31, 2wununi 9073 . . 3  |-  ( ph  ->  U. A  e.  U
)
41, 3wununi 9073 . 2  |-  ( ph  ->  U. U. A  e.  U )
5 ssun2 3654 . . . 4  |-  ran  A  C_  ( dom  A  u.  ran  A )
6 dmrnssfld 5250 . . . 4  |-  ( dom 
A  u.  ran  A
)  C_  U. U. A
75, 6sstri 3498 . . 3  |-  ran  A  C_ 
U. U. A
87a1i 11 . 2  |-  ( ph  ->  ran  A  C_  U. U. A )
91, 4, 8wunss 9079 1  |-  ( ph  ->  ran  A  e.  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823    u. cun 3459    C_ wss 3461   U.cuni 4235   dom cdm 4988   ran crn 4989  WUnicwun 9067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-cnv 4996  df-dm 4998  df-rn 4999  df-wun 9069
This theorem is referenced by:  wuncnv  9097  wunfv  9099  wunco  9100  wuntpos  9101  wunstr  14735  wunfunc  15387  wunnat  15444  catcoppccl  15586  catcfuccl  15587  catcxpccl  15675
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