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Theorem fnotovb 5548
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5215. (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 4376 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B )  -> 
<. C ,  D >.  e.  ( A  X.  B
) )
2 fnopfvb 5215 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  (
( F `  <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F
) )
31, 2sylan2 270 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  ( C  e.  A  /\  D  e.  B
) )  ->  (
( F `  <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F
) )
433impb 1100 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F ) )
5 df-ov 5515 . . 3  |-  ( C F D )  =  ( F `  <. C ,  D >. )
65eqeq1i 2047 . 2  |-  ( ( C F D )  =  R  <->  ( F `  <. C ,  D >. )  =  R )
7 df-ot 3385 . . 3  |-  <. C ,  D ,  R >.  = 
<. <. C ,  D >. ,  R >.
87eleq1i 2103 . 2  |-  ( <. C ,  D ,  R >.  e.  F  <->  <. <. C ,  D >. ,  R >.  e.  F )
94, 6, 83bitr4g 212 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 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   <.cop 3378   <.cotp 3379    X. cxp 4343    Fn wfn 4897   ` cfv 4902  (class class class)co 5512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-ot 3385  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-ov 5515
This theorem is referenced by: (None)
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