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Theorem resoprab 6380
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 5318 . . 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 1951 . . . . 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 2539 . . . . . . . . . 10  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
5 opelxp 5028 . . . . . . . . . 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 704 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  ( A  X.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
87pm5.32i 637 . . . . . . 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 1645 . . . . 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 4511 . . 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 2496 . 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 6325 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
1514reseq1i 5267 . 2  |-  ( {
<. <. x ,  y
>. ,  z >.  | 
ph }  |`  ( A  X.  B ) )  =  ( { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  |`  ( A  X.  B
) )
16 dfoprab2 6325 . 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 2506 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 369    = wceq 1379   E.wex 1596    e. wcel 1767   <.cop 4033   {copab 4504    X. cxp 4997    |` cres 5001   {coprab 6283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005  df-rel 5006  df-res 5011  df-oprab 6286
This theorem is referenced by:  resoprab2  6381  df1stres  27194  df2ndres  27195
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