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Theorem wunrn 8994
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 8971 . . 3  |-  ( ph  ->  U. A  e.  U
)
41, 3wununi 8971 . 2  |-  ( ph  ->  U. U. A  e.  U )
5 ssun2 3615 . . . 4  |-  ran  A  C_  ( dom  A  u.  ran  A )
6 dmrnssfld 5193 . . . 4  |-  ( dom 
A  u.  ran  A
)  C_  U. U. A
75, 6sstri 3460 . . 3  |-  ran  A  C_ 
U. U. A
87a1i 11 . 2  |-  ( ph  ->  ran  A  C_  U. U. A )
91, 4, 8wunss 8977 1  |-  ( ph  ->  ran  A  e.  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758    u. cun 3421    C_ wss 3423   U.cuni 4186   dom cdm 4935   ran crn 4936  WUnicwun 8965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-tr 4481  df-cnv 4943  df-dm 4945  df-rn 4946  df-wun 8967
This theorem is referenced by:  wuncnv  8995  wunfv  8997  wunco  8998  wuntpos  8999  wunstr  14292  wunfunc  14908  wunnat  14965  catcoppccl  15075  catcfuccl  15076  catcxpccl  15116
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