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Theorem rabxp 4914
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 4895 . . . . 5  |-  ( x  e.  ( A  X.  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
21anbi1i 677 . . . 4  |-  ( ( x  e.  ( A  X.  B )  /\  ph )  <->  ( E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) )  /\  ph ) )
3 19.41vv 1925 . . . 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 631 . . . . . 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 685 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( ( ( y  e.  A  /\  z  e.  B )  /\  ph )  <->  ( (
y  e.  A  /\  z  e.  B )  /\  ps ) ) )
7 df-3an 938 . . . . . . . 8  |-  ( ( y  e.  A  /\  z  e.  B  /\  ps )  <->  ( ( y  e.  A  /\  z  e.  B )  /\  ps ) )
86, 7syl6bbr 255 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  ( ( ( y  e.  A  /\  z  e.  B )  /\  ph )  <->  ( y  e.  A  /\  z  e.  B  /\  ps )
) )
98pm5.32i 619 . . . . . 6  |-  ( ( x  =  <. y ,  z >.  /\  (
( y  e.  A  /\  z  e.  B
)  /\  ph ) )  <-> 
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) )
104, 9bitri 241 . . . . 5  |-  ( ( ( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B )
)  /\  ph )  <->  ( x  =  <. y ,  z
>.  /\  ( y  e.  A  /\  z  e.  B  /\  ps )
) )
11102exbii 1593 . . . 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 265 . . 3  |-  ( ( x  e.  ( A  X.  B )  /\  ph )  <->  E. y E. z
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) )
1312abbii 2548 . 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 2714 . 2  |-  { x  e.  ( A  X.  B
)  |  ph }  =  { x  |  ( x  e.  ( A  X.  B )  /\  ph ) }
15 df-opab 4267 . 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 2466 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    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422   {crab 2709   <.cop 3817   {copab 4265    X. cxp 4876
This theorem is referenced by:  fgraphxp  27507  dib1dim  31963  diclspsn  31992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884
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