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Theorem strnfvnd 11995
Description: Deduction version of strnfvn 11996. (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 2699 . 2  |-  ( ph  ->  S  e.  _V )
3 strnfvnd.n . . 3  |-  ( ph  ->  N  e.  NN )
4 fvexg 5440 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( S `  N
)  e.  _V )
51, 3, 4syl2anc 408 . 2  |-  ( ph  ->  ( S `  N
)  e.  _V )
6 fveq1 5420 . . 3  |-  ( x  =  S  ->  (
x `  N )  =  ( S `  N ) )
7 strnfvnd.c . . . 4  |-  E  = Slot 
N
8 df-slot 11979 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
97, 8eqtri 2160 . . 3  |-  E  =  ( x  e.  _V  |->  ( x `  N
) )
106, 9fvmptg 5497 . 2  |-  ( ( S  e.  _V  /\  ( S `  N )  e.  _V )  -> 
( E `  S
)  =  ( S `
 N ) )
112, 5, 10syl2anc 408 1  |-  ( ph  ->  ( E `  S
)  =  ( S `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   _Vcvv 2686    |-> cmpt 3989   ` cfv 5123   NNcn 8734  Slot cslot 11974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-slot 11979
This theorem is referenced by:  strnfvn  11996  strfvssn  11997  strndxid  12003  strsetsid  12008  strslfvd  12016  strslfv2d  12017  setsslid  12025  setsslnid  12026  ressid  12036
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