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Theorem resoprab 6371
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 5308 . . 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 1782 . . . . 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 795 . . . . . . 7  |-  ( ( w  e.  ( A  X.  B )  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) )
4 eleq1 2526 . . . . . . . . . 10  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
5 opelxp 5018 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
64, 5syl6bb 261 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
76anbi1d 702 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  ( A  X.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
87pm5.32i 635 . . . . . . 7  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  ( A  X.  B )  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
93, 8bitri 249 . . . . . 6  |-  ( ( w  e.  ( A  X.  B )  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
1092exbii 1673 . . . . 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 251 . . . 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 4503 . . 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 2483 . 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 6316 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
1514reseq1i 5258 . 2  |-  ( {
<. <. x ,  y
>. ,  z >.  | 
ph }  |`  ( A  X.  B ) )  =  ( { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  |`  ( A  X.  B
) )
16 dfoprab2 6316 . 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 2493 1  |-  ( {
<. <. x ,  y
>. ,  z >.  | 
ph }  |`  ( A  X.  B ) )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  ph ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   <.cop 4022   {copab 4496    X. cxp 4986    |` cres 4990   {coprab 6271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-opab 4498  df-xp 4994  df-rel 4995  df-res 5000  df-oprab 6274
This theorem is referenced by:  resoprab2  6372  df1stres  27750  df2ndres  27751
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