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Theorem fnov 6180
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 5774 . 2  |-  ( F  Fn  ( A  X.  B )  <->  F  =  ( z  e.  ( A  X.  B ) 
|->  ( F `  z
) ) )
2 fveq2 5730 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 6086 . . . . 5  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2488 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54mpt2mpt 6167 . . 3  |-  ( z  e.  ( A  X.  B )  |->  ( F `
 z ) )  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) )
65eqeq2i 2448 . 2  |-  ( F  =  ( z  e.  ( A  X.  B
)  |->  ( F `  z ) )  <->  F  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) ) )
71, 6bitri 242 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 178    = wceq 1653   <.cop 3819    e. cmpt 4268    X. cxp 4878    Fn wfn 5451   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085
This theorem is referenced by:  mapxpen  7275  dfioo2  11007  reschomf  14033  cofulid  14089  cofurid  14090  prf1st  14303  prf2nd  14304  1st2ndprf  14305  curfuncf  14337  curf2ndf  14346  plusfeq  14706  scafeq  15972  psrvscafval  16456  cnfldsub  16731  ipfeq  16883  cnmpt22f  17709  cnmptcom  17712  xkocnv  17848  divstgplem  18152  stdbdxmet  18547  iimulcn  18965  cnnvm  22176  ofpreima  24083  ressplusf  24185  mndpluscn  24314  rmulccn  24316  raddcn  24317  txsconlem  24929  cvmlift2lem6  24997  cvmlift2lem7  24998  cvmlift2lem12  25003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088
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