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Theorem fnov 5952
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 5568 . 2  |-  ( F  Fn  ( A  X.  B )  <->  F  =  ( z  e.  ( A  X.  B ) 
|->  ( F `  z
) ) )
2 fveq2 5525 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5861 . . . . 5  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2333 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54mpt2mpt 5939 . . 3  |-  ( z  e.  ( A  X.  B )  |->  ( F `
 z ) )  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) )
65eqeq2i 2293 . 2  |-  ( F  =  ( z  e.  ( A  X.  B
)  |->  ( F `  z ) )  <->  F  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) ) )
71, 6bitri 240 1  |-  ( F  Fn  ( A  X.  B )  <->  F  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   <.cop 3643    e. cmpt 4077    X. cxp 4687    Fn wfn 5250   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860
This theorem is referenced by:  mapxpen  7027  dfioo2  10744  reschomf  13708  cofulid  13764  cofurid  13765  prf1st  13978  prf2nd  13979  1st2ndprf  13980  curfuncf  14012  curf2ndf  14021  plusfeq  14381  scafeq  15647  psrvscafval  16135  cnfldsub  16402  ipfeq  16554  cnmpt22f  17369  cnmptcom  17372  xkocnv  17505  divstgplem  17803  stdbdxmet  18061  iimulcn  18436  cnnvm  21251  ressplusf  23298  mndpluscn  23299  rmulccn  23301  raddcn  23302  txsconlem  23771  cvmlift2lem6  23839  cvmlift2lem7  23840  cvmlift2lem12  23845  altretop  25600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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