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Theorem fnov 6395
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 5913 . 2  |-  ( F  Fn  ( A  X.  B )  <->  F  =  ( z  e.  ( A  X.  B ) 
|->  ( F `  z
) ) )
2 fveq2 5866 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 6288 . . . . 5  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2526 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54mpt2mpt 6379 . . 3  |-  ( z  e.  ( A  X.  B )  |->  ( F `
 z ) )  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) )
65eqeq2i 2485 . 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 1379   <.cop 4033    |-> cmpt 4505    X. cxp 4997    Fn wfn 5583   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290
This theorem is referenced by:  mapxpen  7684  dfioo2  11626  reschomf  15064  cofulid  15120  cofurid  15121  prf1st  15334  prf2nd  15335  1st2ndprf  15336  curfuncf  15368  curf2ndf  15377  plusfeq  15749  scafeq  17344  psrvscafval  17854  cnfldsub  18257  ipfeq  18492  mdetunilem7  18927  madurid  18953  cnmpt22f  20003  cnmptcom  20006  xkocnv  20142  qustgplem  20446  stdbdxmet  20845  iimulcn  21265  rrxds  21652  rrxmfval  21660  cnnvm  25361  ofpreima  27276  ressplusf  27397  mndpluscn  27659  rmulccn  27661  raddcn  27662  txsconlem  28436  cvmlift2lem6  28504  cvmlift2lem7  28505  cvmlift2lem12  28510
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