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Theorem wunstr 14312
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 9008 . . 3  |-  ( ph  ->  ran  S  e.  U
)
41, 3wununi 8985 . 2  |-  ( ph  ->  U. ran  S  e.  U )
5 ndxarg.1 . . . 4  |-  E  = Slot 
N
65strfvss 14311 . . 3  |-  ( E `
 S )  C_  U.
ran  S
76a1i 11 . 2  |-  ( ph  ->  ( E `  S
)  C_  U. ran  S
)
81, 4, 7wunss 8991 1  |-  ( ph  ->  ( E `  S
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    C_ wss 3437   U.cuni 4200   ran crn 4950   ` cfv 5527  WUnicwun 8979  Slot cslot 14292
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-iota 5490  df-fun 5529  df-fv 5535  df-wun 8981  df-slot 14297
This theorem is referenced by:  wunress  14357  wunfunc  14929  wunnat  14986  catcoppccl  15096  catcfuccl  15097  catcxpccl  15137
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