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Theorem fnov 5968
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 5584 . 2  |-  ( F  Fn  ( A  X.  B )  <->  F  =  ( z  e.  ( A  X.  B ) 
|->  ( F `  z
) ) )
2 fveq2 5541 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5877 . . . . 5  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2346 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54mpt2mpt 5955 . . 3  |-  ( z  e.  ( A  X.  B )  |->  ( F `
 z ) )  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) )
65eqeq2i 2306 . 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 1632   <.cop 3656    e. cmpt 4093    X. cxp 4703    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876
This theorem is referenced by:  mapxpen  7043  dfioo2  10760  reschomf  13724  cofulid  13780  cofurid  13781  prf1st  13994  prf2nd  13995  1st2ndprf  13996  curfuncf  14028  curf2ndf  14037  plusfeq  14397  scafeq  15663  psrvscafval  16151  cnfldsub  16418  ipfeq  16570  cnmpt22f  17385  cnmptcom  17388  xkocnv  17521  divstgplem  17819  stdbdxmet  18077  iimulcn  18452  cnnvm  21267  ressplusf  23313  mndpluscn  23314  rmulccn  23316  raddcn  23317  txsconlem  23786  cvmlift2lem6  23854  cvmlift2lem7  23855  cvmlift2lem12  23860  altretop  25703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879
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