Theorem fnotovb 5965

Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5602. (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 4695	. . . 4 |- ( ( C e. A /\ D e. B ) -> <. C , D >. e. ( A X. B ) )
2		fnopfvb 5602	. . . 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 5925	. . 3 |- ( C F D ) = ( F ` <. C , D >. )
6	5	eqeq1i 2204	. 2 |- ( ( C F D ) = R <-> ( F ` <. C , D >. ) = R )
7		df-ot 3632	. . 3 |- <. C , D , R >. = <. <. C , D >. , R >.
8	7	eleq1i 2262	. 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 2167   <.cop 3625   <.cotp 3626   X. cxp 4661   Fn wfn 5253   ` cfv 5258  (class class class)co 5922

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-ot 3632  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925

This theorem is referenced by: (None)