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Theorem resoprab 6166
Description: Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
Assertion
Ref Expression
resoprab  |-  ( {
<. <. x ,  y
>. ,  z >.  | 
ph }  |`  ( A  X.  B ) )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  ph ) }
Distinct variable groups:    x, y,
z, A    x, B, y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem resoprab
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 resopab 5187 . . 3  |-  ( {
<. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  |`  ( A  X.  B ) )  =  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  E. x E. y ( w  =  <. x ,  y >.  /\  ph ) ) }
2 19.42vv 1930 . . . . 5  |-  ( E. x E. y ( w  e.  ( A  X.  B )  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( w  e.  ( A  X.  B
)  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) ) )
3 an12 773 . . . . . . 7  |-  ( ( w  e.  ( A  X.  B )  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) )
4 eleq1 2496 . . . . . . . . . 10  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
5 opelxp 4908 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
64, 5syl6bb 253 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
76anbi1d 686 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  ( A  X.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
87pm5.32i 619 . . . . . . 7  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  ( A  X.  B )  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
93, 8bitri 241 . . . . . 6  |-  ( ( w  e.  ( A  X.  B )  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
1092exbii 1593 . . . . 5  |-  ( E. x E. y ( w  e.  ( A  X.  B )  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
112, 10bitr3i 243 . . . 4  |-  ( ( w  e.  ( A  X.  B )  /\  E. x E. y ( w  =  <. x ,  y >.  /\  ph ) )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
1211opabbii 4272 . . 3  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  E. x E. y ( w  =  <. x ,  y >.  /\  ph ) ) }  =  { <. w ,  z
>.  |  E. x E. y ( w  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) }
131, 12eqtri 2456 . 2  |-  ( {
<. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  |`  ( A  X.  B ) )  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) }
14 dfoprab2 6121 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
1514reseq1i 5142 . 2  |-  ( {
<. <. x ,  y
>. ,  z >.  | 
ph }  |`  ( A  X.  B ) )  =  ( { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  |`  ( A  X.  B
) )
16 dfoprab2 6121 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ph ) }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) }
1713, 15, 163eqtr4i 2466 1  |-  ( {
<. <. x ,  y
>. ,  z >.  | 
ph }  |`  ( A  X.  B ) )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  ph ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   <.cop 3817   {copab 4265    X. cxp 4876    |` cres 4880   {coprab 6082
This theorem is referenced by:  resoprab2  6167  df1stres  24091  df2ndres  24092
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-ral 2710  df-rex 2711  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  df-rel 4885  df-res 4890  df-oprab 6085
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