MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-oprab Structured version   Unicode version

Definition df-oprab 6085
Description: Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally  x,  y, and  z are distinct, although the definition doesn't strictly require it. See df-ov 6084 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of the most common operation class builder is given by ovmpt2 6209. (Contributed by NM, 12-Mar-1995.)
Assertion
Ref Expression
df-oprab  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
Distinct variable groups:    x, w    y, w    z, w    ph, w
Allowed substitution hints:    ph( x, y, z)

Detailed syntax breakdown of Definition df-oprab
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  set  x
3 vy . . 3  set  y
4 vz . . 3  set  z
51, 2, 3, 4coprab 6082 . 2  class  { <. <.
x ,  y >. ,  z >.  |  ph }
6 vw . . . . . . . . 9  set  w
76cv 1651 . . . . . . . 8  class  w
82cv 1651 . . . . . . . . . 10  class  x
93cv 1651 . . . . . . . . . 10  class  y
108, 9cop 3817 . . . . . . . . 9  class  <. x ,  y >.
114cv 1651 . . . . . . . . 9  class  z
1210, 11cop 3817 . . . . . . . 8  class  <. <. x ,  y >. ,  z
>.
137, 12wceq 1652 . . . . . . 7  wff  w  = 
<. <. x ,  y
>. ,  z >.
1413, 1wa 359 . . . . . 6  wff  ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )
1514, 4wex 1550 . . . . 5  wff  E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph )
1615, 3wex 1550 . . . 4  wff  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
1716, 2wex 1550 . . 3  wff  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )
1817, 6cab 2422 . 2  class  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
195, 18wceq 1652 1  wff  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
Colors of variables: wff set class
This definition is referenced by:  oprabid  6105  dfoprab2  6121  nfoprab1  6123  nfoprab2  6124  nfoprab3  6125  nfoprab  6126  oprabbid  6127  ssoprab2  6130  cbvoprab2  6145  eloprabga  6160  oprabrexex2  6189  eloprabi  6413  mpt20  6427  dftpos3  6497  colinearex  25994
  Copyright terms: Public domain W3C validator