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Theorem resoprab 6379
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 5128 . . 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 1839 . . . . 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 811 . . . . . . 7  |-  ( ( w  e.  ( A  X.  B )  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) )
4 eleq1 2517 . . . . . . . . . 10  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
5 opelxp 4841 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
64, 5syl6bb 269 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
76anbi1d 716 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  ( A  X.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
87pm5.32i 647 . . . . . . 7  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  ( A  X.  B )  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
93, 8bitri 257 . . . . . 6  |-  ( ( w  e.  ( A  X.  B )  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
1092exbii 1722 . . . . 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 259 . . . 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 4438 . . 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 2473 . 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 6324 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
1514reseq1i 5078 . 2  |-  ( {
<. <. x ,  y
>. ,  z >.  | 
ph }  |`  ( A  X.  B ) )  =  ( { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  |`  ( A  X.  B
) )
16 dfoprab2 6324 . 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 2483 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 375    = wceq 1447   E.wex 1666    e. wcel 1890   <.cop 3941   {copab 4431    X. cxp 4809    |` cres 4813   {coprab 6276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pr 4611
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-sn 3936  df-pr 3938  df-op 3942  df-opab 4433  df-xp 4817  df-rel 4818  df-res 4823  df-oprab 6279
This theorem is referenced by:  resoprab2  6380  df1stres  28291  df2ndres  28292
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