Theorem fnotovb 5568

Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5236. (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 4394	. . . 4 |- ( ( C e. A /\ D e. B ) -> <. C , D >. e. ( A X. B ) )
2		fnopfvb 5236	. . . 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 280	. . 3 |- ( ( F Fn ( A X. B ) /\ ( C e. A /\ D e. B ) ) -> ( ( F ` <. C , D >. ) = R <-> <. <. C , D >. , R >. e. F ) )
4	3	3impb 1134	. 2 |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) -> ( ( F ` <. C , D >. ) = R <-> <. <. C , D >. , R >. e. F ) )
5		df-ov 5535	. . 3 |- ( C F D ) = ( F ` <. C , D >. )
6	5	eqeq1i 2088	. 2 |- ( ( C F D ) = R <-> ( F ` <. C , D >. ) = R )
7		df-ot 3408	. . 3 |- <. C , D , R >. = <. <. C , D >. , R >.
8	7	eleq1i 2144	. 2 |- ( <. C , D , R >. e. F <-> <. <. C , D >. , R >. e. F )
9	4, 6, 8	3bitr4g 221	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 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   <.cop 3401   <.cotp 3402   X. cxp 4361   Fn wfn 4917   ` cfv 4922  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-ot 3408  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-ov 5535

This theorem is referenced by: (None)