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Theorem wunstr 14653
Description: Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
ndxarg.1  |-  E  = Slot 
N
wunstr.2  |-  ( ph  ->  U  e. WUni )
wunstr.3  |-  ( ph  ->  S  e.  U )
Assertion
Ref Expression
wunstr  |-  ( ph  ->  ( E `  S
)  e.  U )

Proof of Theorem wunstr
StepHypRef Expression
1 wunstr.2 . 2  |-  ( ph  ->  U  e. WUni )
2 wunstr.3 . . . 4  |-  ( ph  ->  S  e.  U )
31, 2wunrn 9018 . . 3  |-  ( ph  ->  ran  S  e.  U
)
41, 3wununi 8995 . 2  |-  ( ph  ->  U. ran  S  e.  U )
5 ndxarg.1 . . . 4  |-  E  = Slot 
N
65strfvss 14652 . . 3  |-  ( E `
 S )  C_  U.
ran  S
76a1i 11 . 2  |-  ( ph  ->  ( E `  S
)  C_  U. ran  S
)
81, 4, 7wunss 9001 1  |-  ( ph  ->  ( E `  S
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826    C_ wss 3389   U.cuni 4163   ran crn 4914   ` cfv 5496  WUnicwun 8989  Slot cslot 14633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fv 5504  df-wun 8991  df-slot 14638
This theorem is referenced by:  wunress  14701  1strwun  14741  wunfunc  15305  wunnat  15362  catcoppccl  15504  catcfuccl  15505  estrcbasbas  15517  catcxpccl  15593  ringcbasbas  33042  ringcbasbasALTV  33066
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