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Theorem fnotaovb 28039
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5769. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
fnotaovb  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( (( C F D))  =  R  <->  <. C ,  D ,  R >.  e.  F ) )

Proof of Theorem fnotaovb
StepHypRef Expression
1 opelxpi 4911 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B )  -> 
<. C ,  D >.  e.  ( A  X.  B
) )
2 fnopafvb 27996 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  (
( F''' <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F ) )
31, 2sylan2 462 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  ( C  e.  A  /\  D  e.  B
) )  ->  (
( F''' <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F ) )
433impb 1150 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F''' <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F
) )
5 df-aov 27953 . . 3  |- (( C F D))  =  ( F''' <. C ,  D >. )
65eqeq1i 2444 . 2  |-  ( (( C F D))  =  R  <->  ( F''' <. C ,  D >. )  =  R )
7 df-ot 3825 . . 3  |-  <. C ,  D ,  R >.  = 
<. <. C ,  D >. ,  R >.
87eleq1i 2500 . 2  |-  ( <. C ,  D ,  R >.  e.  F  <->  <. <. C ,  D >. ,  R >.  e.  F )
94, 6, 83bitr4g 281 1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( (( C F D))  =  R  <->  <. C ,  D ,  R >.  e.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   <.cop 3818   <.cotp 3819    X. cxp 4877    Fn wfn 5450  '''cafv 27949   ((caov 27950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-ot 3825  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-res 4891  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463  df-dfat 27951  df-afv 27952  df-aov 27953
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