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Theorem rabxp 4890
Description: Membership in a class builder restricted to a Cartesian 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 4870 . . . . 5  |-  ( x  e.  ( A  X.  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
21anbi1i 706 . . . 4  |-  ( ( x  e.  ( A  X.  B )  /\  ph )  <->  ( E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) )  /\  ph ) )
3 19.41vv 1842 . . . 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 659 . . . . . 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 715 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( ( ( y  e.  A  /\  z  e.  B )  /\  ph )  <->  ( (
y  e.  A  /\  z  e.  B )  /\  ps ) ) )
7 df-3an 993 . . . . . . . 8  |-  ( ( y  e.  A  /\  z  e.  B  /\  ps )  <->  ( ( y  e.  A  /\  z  e.  B )  /\  ps ) )
86, 7syl6bbr 271 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  ( ( ( y  e.  A  /\  z  e.  B )  /\  ph )  <->  ( y  e.  A  /\  z  e.  B  /\  ps )
) )
98pm5.32i 647 . . . . . 6  |-  ( ( x  =  <. y ,  z >.  /\  (
( y  e.  A  /\  z  e.  B
)  /\  ph ) )  <-> 
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) )
104, 9bitri 257 . . . . 5  |-  ( ( ( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B )
)  /\  ph )  <->  ( x  =  <. y ,  z
>.  /\  ( y  e.  A  /\  z  e.  B  /\  ps )
) )
11102exbii 1730 . . . 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 281 . . 3  |-  ( ( x  e.  ( A  X.  B )  /\  ph )  <->  E. y E. z
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) )
1312abbii 2578 . 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 2758 . 2  |-  { x  e.  ( A  X.  B
)  |  ph }  =  { x  |  ( x  e.  ( A  X.  B )  /\  ph ) }
15 df-opab 4476 . 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 2494 1  |-  { x  e.  ( A  X.  B
)  |  ph }  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B  /\  ps ) }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898   {cab 2448   {crab 2753   <.cop 3986   {copab 4474    X. cxp 4851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4476  df-xp 4859
This theorem is referenced by:  cicer  15760  poimirlem26  32011  dib1dim  34778  diclspsn  34807  fgraphxp  36133
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