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Theorem rabxp 4463
Description: Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)
Hypothesis
Ref Expression
rabxp.1  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
rabxp  |-  { x  e.  ( A  X.  B
)  |  ph }  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B  /\  ps ) }
Distinct variable groups:    x, y, z, A    x, B, y, z    ph, y, z    ps, x
Allowed substitution hints:    ph( x)    ps( y,
z)

Proof of Theorem rabxp
StepHypRef Expression
1 elxp 4445 . . . . 5  |-  ( x  e.  ( A  X.  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
21anbi1i 446 . . . 4  |-  ( ( x  e.  ( A  X.  B )  /\  ph )  <->  ( E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) )  /\  ph ) )
3 19.41vv 1831 . . . 4  |-  ( E. y E. z ( ( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B )
)  /\  ph )  <->  ( E. y E. z ( x  =  <. y ,  z
>.  /\  ( y  e.  A  /\  z  e.  B ) )  /\  ph ) )
4 anass 393 . . . . . 6  |-  ( ( ( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B )
)  /\  ph )  <->  ( x  =  <. y ,  z
>.  /\  ( ( y  e.  A  /\  z  e.  B )  /\  ph ) ) )
5 rabxp.1 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
65anbi2d 452 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( ( ( y  e.  A  /\  z  e.  B )  /\  ph )  <->  ( (
y  e.  A  /\  z  e.  B )  /\  ps ) ) )
7 df-3an 926 . . . . . . . 8  |-  ( ( y  e.  A  /\  z  e.  B  /\  ps )  <->  ( ( y  e.  A  /\  z  e.  B )  /\  ps ) )
86, 7syl6bbr 196 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  ( ( ( y  e.  A  /\  z  e.  B )  /\  ph )  <->  ( y  e.  A  /\  z  e.  B  /\  ps )
) )
98pm5.32i 442 . . . . . 6  |-  ( ( x  =  <. y ,  z >.  /\  (
( y  e.  A  /\  z  e.  B
)  /\  ph ) )  <-> 
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) )
104, 9bitri 182 . . . . 5  |-  ( ( ( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B )
)  /\  ph )  <->  ( x  =  <. y ,  z
>.  /\  ( y  e.  A  /\  z  e.  B  /\  ps )
) )
11102exbii 1542 . . . 4  |-  ( E. y E. z ( ( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B )
)  /\  ph )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B  /\  ps ) ) )
122, 3, 113bitr2i 206 . . 3  |-  ( ( x  e.  ( A  X.  B )  /\  ph )  <->  E. y E. z
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) )
1312abbii 2203 . 2  |-  { x  |  ( x  e.  ( A  X.  B
)  /\  ph ) }  =  { x  |  E. y E. z
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) }
14 df-rab 2368 . 2  |-  { x  e.  ( A  X.  B
)  |  ph }  =  { x  |  ( x  e.  ( A  X.  B )  /\  ph ) }
15 df-opab 3892 . 2  |-  { <. y ,  z >.  |  ( y  e.  A  /\  z  e.  B  /\  ps ) }  =  {
x  |  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B  /\  ps ) ) }
1613, 14, 153eqtr4i 2118 1  |-  { x  e.  ( A  X.  B
)  |  ph }  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B  /\  ps ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   {crab 2363   <.cop 3444   {copab 3890    X. cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434
This theorem is referenced by: (None)
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