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Theorem strnfvnd 12532
Description: Deduction version of strnfvn 12533. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.)
Hypotheses
Ref Expression
strnfvnd.c  |-  E  = Slot 
N
strnfvnd.f  |-  ( ph  ->  S  e.  V )
strnfvnd.n  |-  ( ph  ->  N  e.  NN )
Assertion
Ref Expression
strnfvnd  |-  ( ph  ->  ( E `  S
)  =  ( S `
 N ) )

Proof of Theorem strnfvnd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 strnfvnd.f . . 3  |-  ( ph  ->  S  e.  V )
21elexd 2765 . 2  |-  ( ph  ->  S  e.  _V )
3 strnfvnd.n . . 3  |-  ( ph  ->  N  e.  NN )
4 fvexg 5553 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( S `  N
)  e.  _V )
51, 3, 4syl2anc 411 . 2  |-  ( ph  ->  ( S `  N
)  e.  _V )
6 fveq1 5533 . . 3  |-  ( x  =  S  ->  (
x `  N )  =  ( S `  N ) )
7 strnfvnd.c . . . 4  |-  E  = Slot 
N
8 df-slot 12516 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
97, 8eqtri 2210 . . 3  |-  E  =  ( x  e.  _V  |->  ( x `  N
) )
106, 9fvmptg 5613 . 2  |-  ( ( S  e.  _V  /\  ( S `  N )  e.  _V )  -> 
( E `  S
)  =  ( S `
 N ) )
112, 5, 10syl2anc 411 1  |-  ( ph  ->  ( E `  S
)  =  ( S `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160   _Vcvv 2752    |-> cmpt 4079   ` cfv 5235   NNcn 8949  Slot cslot 12511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fv 5243  df-slot 12516
This theorem is referenced by:  strnfvn  12533  strfvssn  12534  strndxid  12540  strsetsid  12545  strslfvd  12554  strslfv2d  12555  setsslid  12563  setsslnid  12564  basm  12573
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