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Theorem fnovim 5872
Description: Representation of a function in terms of its values. (Contributed by Jim Kingdon, 16-Jan-2019.)
Assertion
Ref Expression
fnovim  |-  ( 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 fnovim
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dffn5im 5460 . 2  |-  ( F  Fn  ( A  X.  B )  ->  F  =  ( z  e.  ( A  X.  B
)  |->  ( F `  z ) ) )
2 fveq2 5414 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5770 . . . . 5  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2188 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54mpompt 5856 . . 3  |-  ( z  e.  ( A  X.  B )  |->  ( F `
 z ) )  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) )
65eqeq2i 2148 . 2  |-  ( F  =  ( z  e.  ( A  X.  B
)  |->  ( F `  z ) )  <->  F  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) ) )
71, 6sylib 121 1  |-  ( F  Fn  ( A  X.  B )  ->  F  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   <.cop 3525    |-> cmpt 3984    X. cxp 4532    Fn wfn 5113   ` cfv 5118  (class class class)co 5767    e. cmpo 5769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772
This theorem is referenced by:  mapxpen  6735  dfioo2  9750  cnmpt22f  12453  cnmptcom  12456  bdxmet  12659
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