Theorem fnovim 6187

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 5742	. 2 |- ( F Fn ( A X. B ) -> F = ( z e. ( A X. B ) |-> ( F ` z ) ) )
2		fveq2 5690	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 6078	. . . . 5 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2289	. . . 4 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	mpompt 6170	. . 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 ) ) )

Syntax hints:   -> wi 4   = wceq 1402   <.cop 3708   |-> cmpt 4187   X. cxp 4767   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   e. cmpo 6077

This theorem was proved from 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 4244  ax-pow 4306  ax-pr 4341

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080

This theorem is referenced by:  mapxpen  7138  dfioo2  10355  plusfeqg  13661  scafeqg  14617  cnfldadd  14871  cnfldmul  14873  cnfldsub  14884  cnmpt22f  15319  cnmptcom  15322  bdxmet  15525