Theorem fnbrovb 6062

Description: Value of a binary operation expressed as a binary relation. See also fnbrfvb 5684 for functions on Cartesian products. (Contributed by BJ, 15-Feb-2022.)

Assertion

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

Proof of Theorem fnbrovb

Step	Hyp	Ref	Expression
1		df-ov 6020	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
2	1	eqeq1i 2239	. 2 |- ( ( A F B ) = C <-> ( F ` <. A , B >. ) = C )
3		fnbrfvb2 5688	. 2 |- ( ( F Fn ( V X. W ) /\ ( A e. V /\ B e. W ) ) -> ( ( F ` <. A , B >. ) = C <-> <. A , B >. F C ) )
4	2, 3	bitrid 192	1 |- ( ( F Fn ( V X. W ) /\ ( A e. V /\ B e. W ) ) -> ( ( A F B ) = C <-> <. A , B >. F C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   <.cop 3672   class class class wbr 4088   X. cxp 4723   Fn wfn 5321   ` cfv 5326  (class class class)co 6017

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020

This theorem is referenced by: (None)