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Theorem dfoprab2 6154
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 1787 . . . 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 1791 . . . . 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 4080 . . . . . . . . . . . 12  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  z
>.  =  <. <. x ,  y >. ,  z
>. )
43eqeq2d 2454 . . . . . . . . . . 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 1634 . . . . . . 7  |-  ( E. w ( v  = 
<. w ,  z >.  /\  ( w  =  <. x ,  y >.  /\  ph ) )  <->  E. w
( ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  w  =  <. x ,  y
>. ) )
11 opex 4577 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
1211isseti 2999 . . . . . . . 8  |-  E. w  w  =  <. x ,  y >.
13 19.42v 1924 . . . . . . . 8  |-  ( E. w ( ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  w  =  <. x ,  y
>. )  <->  ( ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  /\  E. w  w  =  <. x ,  y >. )
)
1412, 13mpbiran2 910 . . . . . . 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 1636 . . . . 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 1926 . . . . 5  |-  ( E. x E. y ( v  =  <. w ,  z >.  /\  (
w  =  <. x ,  y >.  /\  ph ) )  <->  ( v  =  <. w ,  z
>.  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) ) )
19182exbii 1635 . . . 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 2561 . 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 6116 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { v  |  E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
23 df-opab 4372 . 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 2473 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 1369   E.wex 1586   {cab 2429   <.cop 3904   {copab 4370   {coprab 6113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-opab 4372  df-oprab 6116
This theorem is referenced by:  reloprab  6155  cbvoprab1  6179  cbvoprab12  6181  cbvoprab3  6183  dmoprab  6192  rnoprab  6194  ssoprab2i  6200  mpt2mptx  6202  resoprab  6207  funoprabg  6210  elrnmpt2res  6225  ov6g  6249  dfoprab3s  6650  xpcomco  7422  omxpenlem  7433  nvss  23993  mpt2mptxf  26017  oprabv  30183  mpt2mptx2  30751
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