Theorem fnovim 5887

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 5475	. 2 |- ( F Fn ( A X. B ) -> F = ( z e. ( A X. B ) |-> ( F ` z ) ) )
2		fveq2 5429	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 5785	. . . . 5 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2191	. . . 4 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	mpompt 5871	. . 3 |- ( z e. ( A X. B ) |-> ( F ` z ) ) = ( x e. A , y e. B |-> ( x F y ) )
6	5	eqeq2i 2151	. 2 |- ( F = ( z e. ( A X. B ) |-> ( F ` z ) ) <-> F = ( x e. A , y e. B |-> ( x F y ) ) )
7	1, 6	sylib 121	1 |- ( F Fn ( A X. B ) -> F = ( x e. A , y e. B |-> ( x F y ) ) )

Syntax hints:   -> wi 4   = wceq 1332   <.cop 3535   |-> cmpt 3997   X. cxp 4545   Fn wfn 5126   ` cfv 5131  (class class class)co 5782   e. cmpo 5784

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787

This theorem is referenced by:  mapxpen  6750  dfioo2  9787  cnmpt22f  12503  cnmptcom  12506  bdxmet  12709