ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rabxp Unicode version

Theorem rabxp 4536
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 4516 . . . . 5  |-  ( x  e.  ( A  X.  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
21anbi1i 451 . . . 4  |-  ( ( x  e.  ( A  X.  B )  /\  ph )  <->  ( E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) )  /\  ph ) )
3 19.41vv 1857 . . . 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 396 . . . . . 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 457 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( ( ( y  e.  A  /\  z  e.  B )  /\  ph )  <->  ( (
y  e.  A  /\  z  e.  B )  /\  ps ) ) )
7 df-3an 947 . . . . . . . 8  |-  ( ( y  e.  A  /\  z  e.  B  /\  ps )  <->  ( ( y  e.  A  /\  z  e.  B )  /\  ps ) )
86, 7syl6bbr 197 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  ( ( ( y  e.  A  /\  z  e.  B )  /\  ph )  <->  ( y  e.  A  /\  z  e.  B  /\  ps )
) )
98pm5.32i 447 . . . . . 6  |-  ( ( x  =  <. y ,  z >.  /\  (
( y  e.  A  /\  z  e.  B
)  /\  ph ) )  <-> 
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) )
104, 9bitri 183 . . . . 5  |-  ( ( ( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B )
)  /\  ph )  <->  ( x  =  <. y ,  z
>.  /\  ( y  e.  A  /\  z  e.  B  /\  ps )
) )
11102exbii 1568 . . . 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 207 . . 3  |-  ( ( x  e.  ( A  X.  B )  /\  ph )  <->  E. y E. z
( x  =  <. y ,  z >.  /\  (
y  e.  A  /\  z  e.  B  /\  ps ) ) )
1312abbii 2230 . 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 2399 . 2  |-  { x  e.  ( A  X.  B
)  |  ph }  =  { x  |  ( x  e.  ( A  X.  B )  /\  ph ) }
15 df-opab 3950 . 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 2145 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 103    <-> wb 104    /\ w3a 945    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   {crab 2394   <.cop 3496   {copab 3948    X. cxp 4497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rab 2399  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-opab 3950  df-xp 4505
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator