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Theorem strfvnd 13179
Description: Deduction version of strfvn 13181. (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 2809 . 2  |-  ( S  e.  V  ->  S  e.  _V )
3 fveq1 5540 . . 3  |-  ( x  =  S  ->  (
x `  N )  =  ( S `  N ) )
4 strfvnd.c . . . 4  |-  E  = Slot 
N
5 df-slot 13168 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
64, 5eqtri 2316 . . 3  |-  E  =  ( x  e.  _V  |->  ( x `  N
) )
7 fvex 5555 . . 3  |-  ( S `
 N )  e. 
_V
83, 6, 7fvmpt 5618 . 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 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   ` cfv 5271  Slot cslot 13163
This theorem is referenced by:  strfvn  13181  strfvss  13182  strfvd  13193  strfv2d  13194  setsid  13203  setsnid  13204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-slot 13168
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