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Theorem fnotovb 5908
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5549. (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
StepHypRef Expression
1 opelxpi 4652 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B )  -> 
<. C ,  D >.  e.  ( A  X.  B
) )
2 fnopfvb 5549 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  (
( F `  <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F
) )
31, 2sylan2 286 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  ( C  e.  A  /\  D  e.  B
) )  ->  (
( F `  <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F
) )
433impb 1199 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F ) )
5 df-ov 5868 . . 3  |-  ( C F D )  =  ( F `  <. C ,  D >. )
65eqeq1i 2183 . 2  |-  ( ( C F D )  =  R  <->  ( F `  <. C ,  D >. )  =  R )
7 df-ot 3599 . . 3  |-  <. C ,  D ,  R >.  = 
<. <. C ,  D >. ,  R >.
87eleq1i 2241 . 2  |-  ( <. C ,  D ,  R >.  e.  F  <->  <. <. C ,  D >. ,  R >.  e.  F )
94, 6, 83bitr4g 223 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    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2146   <.cop 3592   <.cotp 3593    X. cxp 4618    Fn wfn 5203   ` cfv 5208  (class class class)co 5865
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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-ot 3599  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-ov 5868
This theorem is referenced by: (None)
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