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Theorem brab2a 4652
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
Hypotheses
Ref Expression
brab2a.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
brab2a.2  |-  R  =  { <. x ,  y
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ph ) }
Assertion
Ref Expression
brab2a  |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y    x, D, y    ps, x, y
Allowed substitution hints:    ph( x, y)    R( x, y)

Proof of Theorem brab2a
StepHypRef Expression
1 simpl 108 . . . . 5  |-  ( ( ( x  e.  C  /\  y  e.  D
)  /\  ph )  -> 
( x  e.  C  /\  y  e.  D
) )
21ssopab2i 4250 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } 
C_  { <. x ,  y >.  |  ( x  e.  C  /\  y  e.  D ) }
3 brab2a.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ph ) }
4 df-xp 4605 . . . 4  |-  ( C  X.  D )  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  e.  D ) }
52, 3, 43sstr4i 3179 . . 3  |-  R  C_  ( C  X.  D
)
65brel 4651 . 2  |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D )
)
7 df-br 3978 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
83eleq2i 2231 . . . 4  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } )
97, 8bitri 183 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } )
10 brab2a.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
1110opelopab2a 4238 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ps ) )
129, 11syl5bb 191 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ps ) )
136, 12biadan2 452 1  |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1342    e. wcel 2135   <.cop 3574   class class class wbr 3977   {copab 4037    X. cxp 4597
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-xp 4605
This theorem is referenced by:  lmbr  12780
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