Theorem fnotovb 5962

Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5599. (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 4692	. . . 4 |- ( ( C e. A /\ D e. B ) -> <. C , D >. e. ( A X. B ) )
2		fnopfvb 5599	. . . 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 1201	. 2 |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) -> ( ( F ` <. C , D >. ) = R <-> <. <. C , D >. , R >. e. F ) )
5		df-ov 5922	. . 3 |- ( C F D ) = ( F ` <. C , D >. )
6	5	eqeq1i 2201	. 2 |- ( ( C F D ) = R <-> ( F ` <. C , D >. ) = R )
7		df-ot 3629	. . 3 |- <. C , D , R >. = <. <. C , D >. , R >.
8	7	eleq1i 2259	. 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 980   = wceq 1364   e. wcel 2164   <.cop 3622   <.cotp 3623   X. cxp 4658   Fn wfn 5250   ` cfv 5255  (class class class)co 5919

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-ot 3629  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922

This theorem is referenced by: (None)