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Theorem strfvnd 13163
Description: Deduction version of strfvn 13165. (Contributed by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
strfvnd.c  |-  E  = Slot 
N
strfvnd.f  |-  ( ph  ->  S  e.  V )
Assertion
Ref Expression
strfvnd  |-  ( ph  ->  ( E `  S
)  =  ( S `
 N ) )

Proof of Theorem strfvnd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 strfvnd.f . 2  |-  ( ph  ->  S  e.  V )
2 elex 2796 . 2  |-  ( S  e.  V  ->  S  e.  _V )
3 fveq1 5524 . . 3  |-  ( x  =  S  ->  (
x `  N )  =  ( S `  N ) )
4 strfvnd.c . . . 4  |-  E  = Slot 
N
5 df-slot 13152 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
64, 5eqtri 2303 . . 3  |-  E  =  ( x  e.  _V  |->  ( x `  N
) )
7 fvex 5539 . . 3  |-  ( S `
 N )  e. 
_V
83, 6, 7fvmpt 5602 . 2  |-  ( S  e.  _V  ->  ( E `  S )  =  ( S `  N ) )
91, 2, 83syl 18 1  |-  ( ph  ->  ( E `  S
)  =  ( S `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   ` cfv 5255  Slot cslot 13147
This theorem is referenced by:  strfvn  13165  strfvss  13166  strfvd  13177  strfv2d  13178  setsid  13187  setsnid  13188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152
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