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Theorem isprmpt2 5889
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 3793 . . . 4  |-  ( F M P  <->  <. F ,  P >.  e.  M )
2 isprmpt2.1 . . . . . 6  |-  ( ph  ->  M  =  { <. f ,  p >.  |  ( f W p  /\  ps ) } )
32adantr 265 . . . . 5  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  ->  M  =  { <. f ,  p >.  |  (
f W p  /\  ps ) } )
43eleq2d 2123 . . . 4  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( <. F ,  P >.  e.  M  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  ( f W p  /\  ps ) } ) )
51, 4syl5bb 185 . . 3  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( F M P  <->  <. F ,  P >.  e. 
{ <. f ,  p >.  |  ( f W p  /\  ps ) } ) )
6 breq12 3797 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f W p  <-> 
F W P ) )
7 isprmpt2.2 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ps  <->  ch )
)
86, 7anbi12d 450 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f W p  /\  ps )  <->  ( F W P  /\  ch ) ) )
98opelopabga 4028 . . . 4  |-  ( ( F  e.  X  /\  P  e.  Y )  ->  ( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f W p  /\  ps ) }  <->  ( F W P  /\  ch )
) )
109adantl 266 . . 3  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f W p  /\  ps ) }  <->  ( F W P  /\  ch )
) )
115, 10bitrd 181 . 2  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( F M P  <-> 
( F W P  /\  ch ) ) )
1211ex 112 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 101    <-> wb 102    = wceq 1259    e. wcel 1409   <.cop 3406   class class class wbr 3792   {copab 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847
This theorem is referenced by: (None)
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