ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  isprmpt2 Unicode version

Theorem isprmpt2 5990
Description: Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Hypotheses
Ref Expression
isprmpt2.1  |-  ( ph  ->  M  =  { <. f ,  p >.  |  ( f W p  /\  ps ) } )
isprmpt2.2  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
isprmpt2  |-  ( ph  ->  ( ( F  e.  X  /\  P  e.  Y )  ->  ( F M P  <->  ( F W P  /\  ch )
) ) )
Distinct variable groups:    f, F, p    P, f, p    f, W, p    ch, f, p
Allowed substitution hints:    ph( f, p)    ps( f, p)    M( f, p)    X( f, p)    Y( f, p)

Proof of Theorem isprmpt2
StepHypRef Expression
1 df-br 3838 . . . 4  |-  ( F M P  <->  <. F ,  P >.  e.  M )
2 isprmpt2.1 . . . . . 6  |-  ( ph  ->  M  =  { <. f ,  p >.  |  ( f W p  /\  ps ) } )
32adantr 270 . . . . 5  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  ->  M  =  { <. f ,  p >.  |  (
f W p  /\  ps ) } )
43eleq2d 2157 . . . 4  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( <. F ,  P >.  e.  M  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  ( f W p  /\  ps ) } ) )
51, 4syl5bb 190 . . 3  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( F M P  <->  <. F ,  P >.  e. 
{ <. f ,  p >.  |  ( f W p  /\  ps ) } ) )
6 breq12 3842 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f W p  <-> 
F W P ) )
7 isprmpt2.2 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ps  <->  ch )
)
86, 7anbi12d 457 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f W p  /\  ps )  <->  ( F W P  /\  ch ) ) )
98opelopabga 4081 . . . 4  |-  ( ( F  e.  X  /\  P  e.  Y )  ->  ( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f W p  /\  ps ) }  <->  ( F W P  /\  ch )
) )
109adantl 271 . . 3  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f W p  /\  ps ) }  <->  ( F W P  /\  ch )
) )
115, 10bitrd 186 . 2  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( F M P  <-> 
( F W P  /\  ch ) ) )
1211ex 113 1  |-  ( ph  ->  ( ( F  e.  X  /\  P  e.  Y )  ->  ( F M P  <->  ( F W P  /\  ch )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   <.cop 3444   class class class wbr 3837   {copab 3890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator