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Theorem dfoprab2 6328
Description: Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)
Assertion
Ref Expression
dfoprab2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
Distinct variable groups:    x, z, w    y, z, w    ph, w
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem dfoprab2
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 excom 1798 . . . 4  |-  ( E. z E. w E. x E. y ( v  =  <. w ,  z
>.  /\  ( w  = 
<. x ,  y >.  /\  ph ) )  <->  E. w E. z E. x E. y ( v  = 
<. w ,  z >.  /\  ( w  =  <. x ,  y >.  /\  ph ) ) )
2 exrot4 1802 . . . . 5  |-  ( E. z E. w E. x E. y ( v  =  <. w ,  z
>.  /\  ( w  = 
<. x ,  y >.  /\  ph ) )  <->  E. x E. y E. z E. w ( v  = 
<. w ,  z >.  /\  ( w  =  <. x ,  y >.  /\  ph ) ) )
3 opeq1 4213 . . . . . . . . . . . 12  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  z
>.  =  <. <. x ,  y >. ,  z
>. )
43eqeq2d 2481 . . . . . . . . . . 11  |-  ( w  =  <. x ,  y
>.  ->  ( v  = 
<. w ,  z >.  <->  v  =  <. <. x ,  y
>. ,  z >. ) )
54pm5.32ri 638 . . . . . . . . . 10  |-  ( ( v  =  <. w ,  z >.  /\  w  =  <. x ,  y
>. )  <->  ( v  = 
<. <. x ,  y
>. ,  z >.  /\  w  =  <. x ,  y >. )
)
65anbi1i 695 . . . . . . . . 9  |-  ( ( ( v  =  <. w ,  z >.  /\  w  =  <. x ,  y
>. )  /\  ph )  <->  ( ( v  =  <. <.
x ,  y >. ,  z >.  /\  w  =  <. x ,  y
>. )  /\  ph )
)
7 anass 649 . . . . . . . . 9  |-  ( ( ( v  =  <. w ,  z >.  /\  w  =  <. x ,  y
>. )  /\  ph )  <->  ( v  =  <. w ,  z >.  /\  (
w  =  <. x ,  y >.  /\  ph ) ) )
8 an32 796 . . . . . . . . 9  |-  ( ( ( v  =  <. <.
x ,  y >. ,  z >.  /\  w  =  <. x ,  y
>. )  /\  ph )  <->  ( ( v  =  <. <.
x ,  y >. ,  z >.  /\  ph )  /\  w  =  <. x ,  y >. )
)
96, 7, 83bitr3i 275 . . . . . . . 8  |-  ( ( v  =  <. w ,  z >.  /\  (
w  =  <. x ,  y >.  /\  ph ) )  <->  ( (
v  =  <. <. x ,  y >. ,  z
>.  /\  ph )  /\  w  =  <. x ,  y >. ) )
109exbii 1644 . . . . . . 7  |-  ( E. w ( v  = 
<. w ,  z >.  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  E. w
( ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  w  =  <. x ,  y
>. ) )
11 opex 4711 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
1211isseti 3119 . . . . . . . 8  |-  E. w  w  =  <. x ,  y >.
13 19.42v 1949 . . . . . . . 8  |-  ( E. w ( ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  w  =  <. x ,  y
>. )  <->  ( ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  E. w  w  =  <. x ,  y >. )
)
1412, 13mpbiran2 917 . . . . . . 7  |-  ( E. w ( ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  w  =  <. x ,  y
>. )  <->  ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
1510, 14bitri 249 . . . . . 6  |-  ( E. w ( v  = 
<. w ,  z >.  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph ) )
16153exbii 1646 . . . . 5  |-  ( E. x E. y E. z E. w ( v  =  <. w ,  z >.  /\  (
w  =  <. x ,  y >.  /\  ph ) )  <->  E. x E. y E. z ( v  =  <. <. x ,  y >. ,  z
>.  /\  ph ) )
172, 16bitri 249 . . . 4  |-  ( E. z E. w E. x E. y ( v  =  <. w ,  z
>.  /\  ( w  = 
<. x ,  y >.  /\  ph ) )  <->  E. x E. y E. z ( v  =  <. <. x ,  y >. ,  z
>.  /\  ph ) )
18 19.42vv 1951 . . . . 5  |-  ( E. x E. y ( v  =  <. w ,  z >.  /\  (
w  =  <. x ,  y >.  /\  ph ) )  <->  ( v  =  <. w ,  z
>.  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) ) )
19182exbii 1645 . . . 4  |-  ( E. w E. z E. x E. y ( v  =  <. w ,  z >.  /\  (
w  =  <. x ,  y >.  /\  ph ) )  <->  E. w E. z ( v  = 
<. w ,  z >.  /\  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) ) )
201, 17, 193bitr3i 275 . . 3  |-  ( E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. w E. z ( v  = 
<. w ,  z >.  /\  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) ) )
2120abbii 2601 . 2  |-  { v  |  E. x E. y E. z ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph ) }  =  { v  |  E. w E. z ( v  =  <. w ,  z
>.  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) ) }
22 df-oprab 6289 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { v  |  E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
23 df-opab 4506 . 2  |-  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  =  { v  |  E. w E. z
( v  =  <. w ,  z >.  /\  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) ) }
2421, 22, 233eqtr4i 2506 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   E.wex 1596   {cab 2452   <.cop 4033   {copab 4504   {coprab 6286
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-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-oprab 6289
This theorem is referenced by:  reloprab  6329  oprabv  6330  cbvoprab1  6354  cbvoprab12  6356  cbvoprab3  6358  dmoprab  6368  rnoprab  6370  ssoprab2i  6376  mpt2mptx  6378  resoprab  6383  funoprabg  6386  elrnmpt2res  6401  ov6g  6425  dfoprab3s  6840  xpcomco  7608  omxpenlem  7619  nvss  25259  mpt2mptxf  27287  mpt2mptx2  32219
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