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Theorem fnotaovb 38571
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5923. (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 4885 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B )  -> 
<. C ,  D >.  e.  ( A  X.  B
) )
2 fnopafvb 38528 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  (
( F''' <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F ) )
31, 2sylan2 476 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  ( C  e.  A  /\  D  e.  B
) )  ->  (
( F''' <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F ) )
433impb 1201 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F''' <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F
) )
5 df-aov 38491 . . 3  |- (( C F D))  =  ( F''' <. C ,  D >. )
65eqeq1i 2429 . 2  |-  ( (( C F D))  =  R  <->  ( F''' <. C ,  D >. )  =  R )
7 df-ot 4007 . . 3  |-  <. C ,  D ,  R >.  = 
<. <. C ,  D >. ,  R >.
87eleq1i 2498 . 2  |-  ( <. C ,  D ,  R >.  e.  F  <->  <. <. C ,  D >. ,  R >.  e.  F )
94, 6, 83bitr4g 291 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   <.cop 4004   <.cotp 4006    X. cxp 4851    Fn wfn 5596  '''cafv 38487   ((caov 38488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-ot 4007  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-res 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-dfat 38489  df-afv 38490  df-aov 38491
This theorem is referenced by: (None)
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