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Theorem fnov 6210
Description: Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
fnov  |-  ( F  Fn  ( A  X.  B )  <->  F  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, F, y

Proof of Theorem fnov
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dffn5 5749 . 2  |-  ( F  Fn  ( A  X.  B )  <->  F  =  ( z  e.  ( A  X.  B ) 
|->  ( F `  z
) ) )
2 fveq2 5703 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 6106 . . . . 5  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2493 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54mpt2mpt 6194 . . 3  |-  ( z  e.  ( A  X.  B )  |->  ( F `
 z ) )  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) )
65eqeq2i 2453 . 2  |-  ( F  =  ( z  e.  ( A  X.  B
)  |->  ( F `  z ) )  <->  F  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) ) )
71, 6bitri 249 1  |-  ( F  Fn  ( A  X.  B )  <->  F  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369   <.cop 3895    e. cmpt 4362    X. cxp 4850    Fn wfn 5425   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fn 5433  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108
This theorem is referenced by:  mapxpen  7489  dfioo2  11402  reschomf  14756  cofulid  14812  cofurid  14813  prf1st  15026  prf2nd  15027  1st2ndprf  15028  curfuncf  15060  curf2ndf  15069  plusfeq  15441  scafeq  16980  psrvscafval  17473  cnfldsub  17856  ipfeq  18091  mdetunilem7  18436  madurid  18462  cnmpt22f  19260  cnmptcom  19263  xkocnv  19399  divstgplem  19703  stdbdxmet  20102  iimulcn  20522  rrxds  20909  rrxmfval  20917  cnnvm  24085  ofpreima  25996  ressplusf  26123  mndpluscn  26368  rmulccn  26370  raddcn  26371  txsconlem  27141  cvmlift2lem6  27209  cvmlift2lem7  27210  cvmlift2lem12  27215
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