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Theorem isprmpt2 6979
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 4427 . . . 4  |-  ( F M P  <->  <. F ,  P >.  e.  M )
2 isprmpt2.1 . . . . . 6  |-  ( ph  ->  M  =  { <. f ,  p >.  |  ( f W p  /\  ps ) } )
32adantr 466 . . . . 5  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  ->  M  =  { <. f ,  p >.  |  (
f W p  /\  ps ) } )
43eleq2d 2499 . . . 4  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( <. F ,  P >.  e.  M  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  ( f W p  /\  ps ) } ) )
51, 4syl5bb 260 . . 3  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( F M P  <->  <. F ,  P >.  e. 
{ <. f ,  p >.  |  ( f W p  /\  ps ) } ) )
6 breq12 4431 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f W p  <-> 
F W P ) )
7 isprmpt2.2 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ps  <->  ch )
)
86, 7anbi12d 715 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f W p  /\  ps )  <->  ( F W P  /\  ch ) ) )
98opelopabga 4734 . . . 4  |-  ( ( F  e.  X  /\  P  e.  Y )  ->  ( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f W p  /\  ps ) }  <->  ( F W P  /\  ch )
) )
109adantl 467 . . 3  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f W p  /\  ps ) }  <->  ( F W P  /\  ch )
) )
115, 10bitrd 256 . 2  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( F M P  <-> 
( F W P  /\  ch ) ) )
1211ex 435 1  |-  ( ph  ->  ( ( F  e.  X  /\  P  e.  Y )  ->  ( F M P  <->  ( F W P  /\  ch )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   <.cop 4008   class class class wbr 4426   {copab 4483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485
This theorem is referenced by:  iscrct  25197  iscycl  25198
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