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Theorem strfvnd 13411
Description: Deduction version of strfvn 13413. (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 2907 . 2  |-  ( S  e.  V  ->  S  e.  _V )
3 fveq1 5667 . . 3  |-  ( x  =  S  ->  (
x `  N )  =  ( S `  N ) )
4 strfvnd.c . . . 4  |-  E  = Slot 
N
5 df-slot 13400 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
64, 5eqtri 2407 . . 3  |-  E  =  ( x  e.  _V  |->  ( x `  N
) )
7 fvex 5682 . . 3  |-  ( S `
 N )  e. 
_V
83, 6, 7fvmpt 5745 . 2  |-  ( S  e.  _V  ->  ( E `  S )  =  ( S `  N ) )
91, 2, 83syl 19 1  |-  ( ph  ->  ( E `  S
)  =  ( S `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2899    e. cmpt 4207   ` cfv 5394  Slot cslot 13395
This theorem is referenced by:  strfvn  13413  strfvss  13414  strfvd  13425  strfv2d  13426  setsid  13435  setsnid  13436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-slot 13400
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