Theorem fnovim 6197

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.

Step	Hyp	Ref	Expression
1		dffn5im 5748	. 2 |- ( F Fn ( A X. B ) -> F = ( z e. ( A X. B ) |-> ( F ` z ) ) )
2		fveq2 5695	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 6088	. . . . 5 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2289	. . . 4 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	mpompt 6180	. . 3 |- ( z e. ( A X. B ) |-> ( F ` z ) ) = ( x e. A , y e. B |-> ( x F y ) )
6	5	eqeq2i 2249	. 2 |- ( F = ( z e. ( A X. B ) |-> ( F ` z ) ) <-> F = ( x e. A , y e. B |-> ( x F y ) ) )
7	1, 6	sylib 122	1 |- ( F Fn ( A X. B ) -> F = ( x e. A , y e. B |-> ( x F y ) ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   <.cop 3712   |-> cmpt 4192   X. cxp 4772   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   e. cmpo 6087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090

This theorem is used by:  mapxpen  7148  dfioo2  10376  plusfeqg  13684  scafeqg  14645  cnfldadd  14899  cnfldmul  14901  cnfldsub  14912  cnmpt22f  15396  cnmptcom  15399  bdxmet  15602