Theorem fnovim 5879

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 5467	. 2 |- ( F Fn ( A X. B ) -> F = ( z e. ( A X. B ) |-> ( F ` z ) ) )
2		fveq2 5421	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 5777	. . . . 5 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	syl6eqr 2190	. . . 4 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	mpompt 5863	. . 3 |- ( z e. ( A X. B ) |-> ( F ` z ) ) = ( x e. A , y e. B |-> ( x F y ) )
6	5	eqeq2i 2150	. 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 1331   <.cop 3530   |-> cmpt 3989   X. cxp 4537   Fn wfn 5118   ` cfv 5123  (class class class)co 5774   e. cmpo 5776

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779

This theorem is referenced by:  mapxpen  6742  dfioo2  9757  cnmpt22f  12464  cnmptcom  12467  bdxmet  12670