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Theorem rbropapd 6221
Description: Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Hypotheses
Ref Expression
rbropapd.1  |-  ( ph  ->  M  =  { <. f ,  p >.  |  ( f W p  /\  ps ) } )
rbropapd.2  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
rbropapd  |-  ( 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 rbropapd
StepHypRef Expression
1 df-br 3990 . . . 4  |-  ( F M P  <->  <. F ,  P >.  e.  M )
2 rbropapd.1 . . . . 5  |-  ( ph  ->  M  =  { <. f ,  p >.  |  ( f W p  /\  ps ) } )
32eleq2d 2240 . . . 4  |-  ( ph  ->  ( <. F ,  P >.  e.  M  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  ( f W p  /\  ps ) } ) )
41, 3syl5bb 191 . . 3  |-  ( ph  ->  ( F M P  <->  <. F ,  P >.  e. 
{ <. f ,  p >.  |  ( f W p  /\  ps ) } ) )
5 breq12 3994 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f W p  <-> 
F W P ) )
6 rbropapd.2 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ps  <->  ch )
)
75, 6anbi12d 470 . . . 4  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f W p  /\  ps )  <->  ( F W P  /\  ch ) ) )
87opelopabga 4248 . . 3  |-  ( ( F  e.  X  /\  P  e.  Y )  ->  ( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f W p  /\  ps ) }  <->  ( F W P  /\  ch )
) )
94, 8sylan9bb 459 . 2  |-  ( (
ph  /\  ( F  e.  X  /\  P  e.  Y ) )  -> 
( F M P  <-> 
( F W P  /\  ch ) ) )
109ex 114 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 103    <-> wb 104    = wceq 1348    e. wcel 2141   <.cop 3586   class class class wbr 3989   {copab 4049
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051
This theorem is referenced by:  rbropap  6222
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