Theorem fnotovb 5885

Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5528. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fnotovb	|- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) -> ( ( C F D ) = R <-> <. C , D , R >. e. F ) )

Proof of Theorem fnotovb

Step	Hyp	Ref	Expression
1		opelxpi 4636	. . . 4 |- ( ( C e. A /\ D e. B ) -> <. C , D >. e. ( A X. B ) )
2		fnopfvb 5528	. . . 4 |- ( ( F Fn ( A X. B ) /\ <. C , D >. e. ( A X. B ) ) -> ( ( F ` <. C , D >. ) = R <-> <. <. C , D >. , R >. e. F ) )
3	1, 2	sylan2 284	. . 3 |- ( ( F Fn ( A X. B ) /\ ( C e. A /\ D e. B ) ) -> ( ( F ` <. C , D >. ) = R <-> <. <. C , D >. , R >. e. F ) )
4	3	3impb 1189	. 2 |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) -> ( ( F ` <. C , D >. ) = R <-> <. <. C , D >. , R >. e. F ) )
5		df-ov 5845	. . 3 |- ( C F D ) = ( F ` <. C , D >. )
6	5	eqeq1i 2173	. 2 |- ( ( C F D ) = R <-> ( F ` <. C , D >. ) = R )
7		df-ot 3586	. . 3 |- <. C , D , R >. = <. <. C , D >. , R >.
8	7	eleq1i 2232	. 2 |- ( <. C , D , R >. e. F <-> <. <. C , D >. , R >. e. F )
9	4, 6, 8	3bitr4g 222	1 |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) -> ( ( C F D ) = R <-> <. C , D , R >. e. F ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   <.cop 3579   <.cotp 3580   X. cxp 4602   Fn wfn 5183   ` cfv 5188  (class class class)co 5842

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-ot 3586  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196  df-ov 5845

This theorem is referenced by: (None)