Theorem fnotoprb 4916

Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 4713.

Hypothesis

Ref	Expression
fnotoprb.1	|- R e. _V

Assertion

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

Proof of Theorem fnotoprb

Step	Hyp	Ref	Expression
1		fnotoprb.1	. . . . 5 |- R e. _V
2	1	fnopfvb 4713	. . . 4 |- ((F Fn (A X. B) /\ <.C, D>. e. (A X. B)) -> ((F` <.C, D>.) = R <-> <.<.C, D>., R>. e. F))
3		df-opr 4886	. . . . 5 |- (CFD) = (F` <.C, D>.)
4	3	eqeq1i 1891	. . . 4 |- ((CFD) = R <-> (F` <.C, D>.) = R)
5	2, 4	syl5bb 591	. . 3 |- ((F Fn (A X. B) /\ <.C, D>. e. (A X. B)) -> ((CFD) = R <-> <.<.C, D>., R>. e. F))
6		opelxpi 4040	. . 3 |- ((C e. A /\ D e. B) -> <.C, D>. e. (A X. B))
7	5, 6	sylan2 500	. 2 |- ((F Fn (A X. B) /\ (C e. A /\ D e. B)) -> ((CFD) = R <-> <.<.C, D>., R>. e. F))
8	7	3impb 1063	1 |- ((F Fn (A X. B) /\ C e. A /\ D e. B) -> ((CFD) = R <-> <.<.C, D>., R>. e. F))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046   X. cxp 3984   Fn wfn 3993  ` cfv 3998  (class class class)co 4884

This theorem is referenced by:  zrdivrng 10418  prj1 14395

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886

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