ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnotovb Unicode version

Theorem fnotovb 5807
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5456. (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 4566 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B )  -> 
<. C ,  D >.  e.  ( A  X.  B
) )
2 fnopfvb 5456 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  (
( F `  <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F
) )
31, 2sylan2 284 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  ( C  e.  A  /\  D  e.  B
) )  ->  (
( F `  <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F
) )
433impb 1177 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  <. C ,  D >. )  =  R  <->  <. <. C ,  D >. ,  R >.  e.  F ) )
5 df-ov 5770 . . 3  |-  ( C F D )  =  ( F `  <. C ,  D >. )
65eqeq1i 2145 . 2  |-  ( ( C F D )  =  R  <->  ( F `  <. C ,  D >. )  =  R )
7 df-ot 3532 . . 3  |-  <. C ,  D ,  R >.  = 
<. <. C ,  D >. ,  R >.
87eleq1i 2203 . 2  |-  ( <. C ,  D ,  R >.  e.  F  <->  <. <. C ,  D >. ,  R >.  e.  F )
94, 6, 83bitr4g 222 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 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   <.cop 3525   <.cotp 3526    X. cxp 4532    Fn wfn 5113   ` cfv 5118  (class class class)co 5767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-ot 3532  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126  df-ov 5770
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator