Theorem fnotovb 6074

Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5694. (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 4763	. . . 4 |- ( ( C e. A /\ D e. B ) -> <. C , D >. e. ( A X. B ) )
2		fnopfvb 5694	. . . 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 286	. . 3 |- ( ( F Fn ( A X. B ) /\ ( C e. A /\ D e. B ) ) -> ( ( F ` <. C , D >. ) = R <-> <. <. C , D >. , R >. e. F ) )
4	3	3impb 1226	. 2 |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) -> ( ( F ` <. C , D >. ) = R <-> <. <. C , D >. , R >. e. F ) )
5		df-ov 6031	. . 3 |- ( C F D ) = ( F ` <. C , D >. )
6	5	eqeq1i 2239	. 2 |- ( ( C F D ) = R <-> ( F ` <. C , D >. ) = R )
7		df-ot 3683	. . 3 |- <. C , D , R >. = <. <. C , D >. , R >.
8	7	eleq1i 2297	. 2 |- ( <. C , D , R >. e. F <-> <. <. C , D >. , R >. e. F )
9	4, 6, 8	3bitr4g 223	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 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   <.cop 3676   <.cotp 3677   X. cxp 4729   Fn wfn 5328   ` cfv 5333  (class class class)co 6028

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-ot 3683  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031

This theorem is referenced by: (None)