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Theorem dfoprab2 5911
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 1831 . . . . 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 3812 . . . . . . . . . . . 12  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  z
>.  =  <. <. x ,  y >. ,  z
>. )
43eqeq2d 2307 . . . . . . . . . . 11  |-  ( w  =  <. x ,  y
>.  ->  ( v  = 
<. w ,  z >.  <->  v  =  <. <. x ,  y
>. ,  z >. ) )
54pm5.32ri 619 . . . . . . . . . 10  |-  ( ( v  =  <. w ,  z >.  /\  w  =  <. x ,  y
>. )  <->  ( v  = 
<. <. x ,  y
>. ,  z >.  /\  w  =  <. x ,  y >. )
)
65anbi1i 676 . . . . . . . . 9  |-  ( ( ( v  =  <. w ,  z >.  /\  w  =  <. x ,  y
>. )  /\  ph )  <->  ( ( v  =  <. <.
x ,  y >. ,  z >.  /\  w  =  <. x ,  y
>. )  /\  ph )
)
7 anass 630 . . . . . . . . 9  |-  ( ( ( v  =  <. w ,  z >.  /\  w  =  <. x ,  y
>. )  /\  ph )  <->  ( v  =  <. w ,  z >.  /\  (
w  =  <. x ,  y >.  /\  ph ) ) )
8 an32 773 . . . . . . . . 9  |-  ( ( ( v  =  <. <.
x ,  y >. ,  z >.  /\  w  =  <. x ,  y
>. )  /\  ph )  <->  ( ( v  =  <. <.
x ,  y >. ,  z >.  /\  ph )  /\  w  =  <. x ,  y >. )
)
96, 7, 83bitr3i 266 . . . . . . . 8  |-  ( ( v  =  <. w ,  z >.  /\  (
w  =  <. x ,  y >.  /\  ph ) )  <->  ( (
v  =  <. <. x ,  y >. ,  z
>.  /\  ph )  /\  w  =  <. x ,  y >. ) )
109exbii 1572 . . . . . . 7  |-  ( E. w ( v  = 
<. w ,  z >.  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  E. w
( ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  w  =  <. x ,  y
>. ) )
11 opex 4253 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
1211isseti 2807 . . . . . . . 8  |-  E. w  w  =  <. x ,  y >.
13 19.42v 1858 . . . . . . . 8  |-  ( E. w ( ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  w  =  <. x ,  y
>. )  <->  ( ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  E. w  w  =  <. x ,  y >. )
)
1412, 13mpbiran2 885 . . . . . . 7  |-  ( E. w ( ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  w  =  <. x ,  y
>. )  <->  ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
1510, 14bitri 240 . . . . . 6  |-  ( E. w ( v  = 
<. w ,  z >.  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph ) )
16153exbii 1574 . . . . 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 240 . . . 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 1860 . . . . 5  |-  ( E. x E. y ( v  =  <. w ,  z >.  /\  (
w  =  <. x ,  y >.  /\  ph ) )  <->  ( v  =  <. w ,  z
>.  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) ) )
19182exbii 1573 . . . 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 266 . . 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 2408 . 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 5878 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { v  |  E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
23 df-opab 4094 . 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 2326 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632   {cab 2282   <.cop 3656   {copab 4092   {coprab 5875
This theorem is referenced by:  reloprab  5912  cbvoprab1  5934  cbvoprab12  5936  cbvoprab3  5938  dmoprab  5944  rnoprab  5946  ssoprab2i  5952  mpt2mptx  5954  resoprab  5956  funoprabg  5959  ov6g  6001  dfoprab3s  6191  xpcomco  6968  omxpenlem  6979  nvss  21165  ssoprab2g  25135  dfoprab4pop  25140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-oprab 5878
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