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Theorem resoprab 5625
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 4680 . . 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 1804 . . . . 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 503 . . . . . . 7  |-  ( ( w  e.  ( A  X.  B )  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) )
4 eleq1 2116 . . . . . . . . . 10  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
5 opelxp 4402 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
64, 5syl6bb 189 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
76anbi1d 446 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  ( A  X.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
87pm5.32i 435 . . . . . . 7  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  ( A  X.  B )  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
93, 8bitri 177 . . . . . 6  |-  ( ( w  e.  ( A  X.  B )  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
1092exbii 1513 . . . . 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 179 . . . 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 3852 . . 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 2076 . 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 5580 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
1514reseq1i 4636 . 2  |-  ( {
<. <. x ,  y
>. ,  z >.  | 
ph }  |`  ( A  X.  B ) )  =  ( { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  |`  ( A  X.  B
) )
16 dfoprab2 5580 . 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 2086 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 101    = wceq 1259   E.wex 1397    e. wcel 1409   <.cop 3406   {copab 3845    X. cxp 4371    |` cres 4375   {coprab 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379  df-rel 4380  df-res 4385  df-oprab 5544
This theorem is referenced by:  resoprab2  5626
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