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

Theorem reloprab 6054
Description: An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
Assertion
Ref Expression
reloprab  |-  Rel  { <. <. x ,  y
>. ,  z >.  | 
ph }
Distinct variable groups:    x, z    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem reloprab
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dfoprab2 6053 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
21relopabi 4933 1  |-  Rel  { <. <. x ,  y
>. ,  z >.  | 
ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649   <.cop 3753   Rel wrel 4816   {coprab 6014
This theorem is referenced by:  oprabss  6091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-opab 4201  df-xp 4817  df-rel 4818  df-oprab 6017
  Copyright terms: Public domain W3C validator