ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  strnfvnd Unicode version

Theorem strnfvnd 11761
Description: Deduction version of strnfvn 11762. (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 2654 . 2  |-  ( ph  ->  S  e.  _V )
3 strnfvnd.n . . 3  |-  ( ph  ->  N  e.  NN )
4 fvexg 5372 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( S `  N
)  e.  _V )
51, 3, 4syl2anc 406 . 2  |-  ( ph  ->  ( S `  N
)  e.  _V )
6 fveq1 5352 . . 3  |-  ( x  =  S  ->  (
x `  N )  =  ( S `  N ) )
7 strnfvnd.c . . . 4  |-  E  = Slot 
N
8 df-slot 11745 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
97, 8eqtri 2120 . . 3  |-  E  =  ( x  e.  _V  |->  ( x `  N
) )
106, 9fvmptg 5429 . 2  |-  ( ( S  e.  _V  /\  ( S `  N )  e.  _V )  -> 
( E `  S
)  =  ( S `
 N ) )
112, 5, 10syl2anc 406 1  |-  ( ph  ->  ( E `  S
)  =  ( S `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1299    e. wcel 1448   _Vcvv 2641    |-> cmpt 3929   ` cfv 5059   NNcn 8578  Slot cslot 11740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fv 5067  df-slot 11745
This theorem is referenced by:  strnfvn  11762  strfvssn  11763  strndxid  11769  strsetsid  11774  strslfvd  11782  strslfv2d  11783  setsslid  11791  setsslnid  11792  ressid  11802
  Copyright terms: Public domain W3C validator