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Theorem isprmpt2 6953
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 4448 . . . 4  |-  ( F M P  <->  <. F ,  P >.  e.  M )
2 isprmpt2.1 . . . . . 6  |-  ( ph  ->  M  =  { <. f ,  p >.  |  ( f W p  /\  ps ) } )
32adantr 465 . . . . 5  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  ->  M  =  { <. f ,  p >.  |  (
f W p  /\  ps ) } )
43eleq2d 2537 . . . 4  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( <. F ,  P >.  e.  M  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  ( f W p  /\  ps ) } ) )
51, 4syl5bb 257 . . 3  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( F M P  <->  <. F ,  P >.  e. 
{ <. f ,  p >.  |  ( f W p  /\  ps ) } ) )
6 breq12 4452 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f W p  <-> 
F W P ) )
7 isprmpt2.2 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ps  <->  ch )
)
86, 7anbi12d 710 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f W p  /\  ps )  <->  ( F W P  /\  ch ) ) )
98opelopabga 4760 . . . 4  |-  ( ( F  e.  X  /\  P  e.  Y )  ->  ( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f W p  /\  ps ) }  <->  ( F W P  /\  ch )
) )
109adantl 466 . . 3  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f W p  /\  ps ) }  <->  ( F W P  /\  ch )
) )
115, 10bitrd 253 . 2  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( F M P  <-> 
( F W P  /\  ch ) ) )
1211ex 434 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447   {copab 4504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506
This theorem is referenced by:  iscrct  24316  iscycl  24317
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