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Theorem rabxp 4725
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 4706 . . . . 5  |-  ( x  e.  ( A  X.  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
21anbi1i 676 . . . 4  |-  ( ( x  e.  ( A  X.  B )  /\  ph )  <->  ( E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) )  /\  ph ) )
3 19.41vv 1843 . . . 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 630 . . . . . 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 684 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( ( ( y  e.  A  /\  z  e.  B )  /\  ph )  <->  ( (
y  e.  A  /\  z  e.  B )  /\  ps ) ) )
7 df-3an 936 . . . . . . . 8  |-  ( ( y  e.  A  /\  z  e.  B  /\  ps )  <->  ( ( y  e.  A  /\  z  e.  B )  /\  ps ) )
86, 7syl6bbr 254 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  ( ( ( y  e.  A  /\  z  e.  B )  /\  ph )  <->  ( y  e.  A  /\  z  e.  B  /\  ps )
) )
98pm5.32i 618 . . . . . 6  |-  ( ( x  =  <. y ,  z >.  /\  (
( y  e.  A  /\  z  e.  B
)  /\  ph ) )  <-> 
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) )
104, 9bitri 240 . . . . 5  |-  ( ( ( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B )
)  /\  ph )  <->  ( x  =  <. y ,  z
>.  /\  ( y  e.  A  /\  z  e.  B  /\  ps )
) )
11102exbii 1570 . . . 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 264 . . 3  |-  ( ( x  e.  ( A  X.  B )  /\  ph )  <->  E. y E. z
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) )
1312abbii 2395 . 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 2552 . 2  |-  { x  e.  ( A  X.  B
)  |  ph }  =  { x  |  ( x  e.  ( A  X.  B )  /\  ph ) }
15 df-opab 4078 . 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 2313 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 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547   <.cop 3643   {copab 4076    X. cxp 4687
This theorem is referenced by:  fgraphxp  26942  dib1dim  30728  diclspsn  30757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695
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