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Theorem brab2a 4754
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 443 . . . . 5  |-  ( ( ( x  e.  C  /\  y  e.  D
)  /\  ph )  -> 
( x  e.  C  /\  y  e.  D
) )
21ssopab2i 4308 . . . 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 4711 . . . 4  |-  ( C  X.  D )  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  e.  D ) }
52, 3, 43sstr4i 3230 . . 3  |-  R  C_  ( C  X.  D
)
65brel 4753 . 2  |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D )
)
7 df-br 4040 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
83eleq2i 2360 . . . 4  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } )
97, 8bitri 240 . . 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 4296 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ps ) )
129, 11syl5bb 248 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ps ) )
136, 12biadan2 623 1  |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   {copab 4092    X. cxp 4703
This theorem is referenced by:  issect  13672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711
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