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

Definition df-oprab 5878
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 5877 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 5999. (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 5875 . 2  class  { <. <.
x ,  y >. ,  z >.  |  ph }
6 vw . . . . . . . . 9  set  w
76cv 1631 . . . . . . . 8  class  w
82cv 1631 . . . . . . . . . 10  class  x
93cv 1631 . . . . . . . . . 10  class  y
108, 9cop 3656 . . . . . . . . 9  class  <. x ,  y >.
114cv 1631 . . . . . . . . 9  class  z
1210, 11cop 3656 . . . . . . . 8  class  <. <. x ,  y >. ,  z
>.
137, 12wceq 1632 . . . . . . 7  wff  w  = 
<. <. x ,  y
>. ,  z >.
1413, 1wa 358 . . . . . 6  wff  ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )
1514, 4wex 1531 . . . . 5  wff  E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph )
1615, 3wex 1531 . . . 4  wff  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
1716, 2wex 1531 . . 3  wff  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )
1817, 6cab 2282 . 2  class  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
195, 18wceq 1632 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  5898  dfoprab2  5911  nfoprab1  5913  nfoprab2  5914  nfoprab3  5915  nfoprab  5916  oprabbid  5917  ssoprab2  5920  cbvoprab2  5935  eloprabga  5950  oprabrexex2  5979  eloprabi  6202  mpt20  6215  dftpos3  6268  colinearex  24755
  Copyright terms: Public domain W3C validator