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

Theorem dmoprab 5944
Description: The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Assertion
Ref Expression
dmoprab  |-  dom  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  { <. x ,  y >.  |  E. z ph }
Distinct variable groups:    x, z    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem dmoprab
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dfoprab2 5911 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
21dmeqi 4896 . 2  |-  dom  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  dom  {
<. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
3 dmopab 4905 . 2  |-  dom  { <. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. z E. x E. y ( w  =  <. x ,  y >.  /\  ph ) }
4 exrot3 1830 . . . . 5  |-  ( E. z E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. z ( w  =  <. x ,  y >.  /\  ph ) )
5 19.42v 1858 . . . . . 6  |-  ( E. z ( w  = 
<. x ,  y >.  /\  ph )  <->  ( w  =  <. x ,  y
>.  /\  E. z ph ) )
652exbii 1573 . . . . 5  |-  ( E. x E. y E. z ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  E. z ph )
)
74, 6bitri 240 . . . 4  |-  ( E. z E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  E. z ph )
)
87abbii 2408 . . 3  |-  { w  |  E. z E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) }  =  { w  |  E. x E. y ( w  =  <. x ,  y
>.  /\  E. z ph ) }
9 df-opab 4094 . . 3  |-  { <. x ,  y >.  |  E. z ph }  =  {
w  |  E. x E. y ( w  = 
<. x ,  y >.  /\  E. z ph ) }
108, 9eqtr4i 2319 . 2  |-  { w  |  E. z E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) }  =  { <. x ,  y
>.  |  E. z ph }
112, 3, 103eqtri 2320 1  |-  dom  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  { <. x ,  y >.  |  E. z ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632   {cab 2282   <.cop 3656   {copab 4092   dom cdm 4705   {coprab 5875
This theorem is referenced by:  dmoprabss  5945  reldmoprab  5948  fnoprabg  5961  1st2val  6161  2nd2val  6162  linedegen  24838  dmoprabss6  25138  cmppar2  26075
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-eu 2160  df-mo 2161  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-br 4040  df-opab 4094  df-dm 4715  df-oprab 5878
  Copyright terms: Public domain W3C validator