Theorem fnbrfvb2 5721

Description: Version of fnbrfvb 5717 for functions on Cartesian products: function value expressed as a binary relation. See fnbrovb 6097 for the form when F is seen as a binary operation. (Contributed by BJ, 15-Feb-2022.)

Assertion

Ref	Expression
fnbrfvb2	|- ( ( F Fn ( V X. W ) /\ ( A e. V /\ B e. W ) ) -> ( ( F ` <. A , B >. ) = C <-> <. A , B >. F C ) )

Proof of Theorem fnbrfvb2

Step	Hyp	Ref	Expression
1		opelxpi 4783	. 2 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. ( V X. W ) )
2		fnbrfvb 5717	. 2 |- ( ( F Fn ( V X. W ) /\ <. A , B >. e. ( V X. W ) ) -> ( ( F ` <. A , B >. ) = C <-> <. A , B >. F C ) )
3	1, 2	sylan2 286	1 |- ( ( F Fn ( V X. W ) /\ ( A e. V /\ B e. W ) ) -> ( ( F ` <. A , B >. ) = C <-> <. A , B >. F C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   <.cop 3694   class class class wbr 4111   X. cxp 4749   Fn wfn 5349   ` cfv 5354

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362

This theorem is referenced by:  fnbrovb  6097