Theorem fnovim 6104

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 5672	. 2 |- ( F Fn ( A X. B ) -> F = ( z e. ( A X. B ) |-> ( F ` z ) ) )
2		fveq2 5623	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 5997	. . . . 5 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2280	. . . 4 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	mpompt 6087	. . 3 |- ( z e. ( A X. B ) |-> ( F ` z ) ) = ( x e. A , y e. B |-> ( x F y ) )
6	5	eqeq2i 2240	. 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 1395   <.cop 3669   |-> cmpt 4144   X. cxp 4714   Fn wfn 5309   ` cfv 5314  (class class class)co 5994   e. cmpo 5996

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999

This theorem is referenced by:  mapxpen  6997  dfioo2  10158  plusfeqg  13383  scafeqg  14257  cnfldadd  14511  cnfldmul  14513  cnfldsub  14524  cnmpt22f  14954  cnmptcom  14957  bdxmet  15160