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Theorem dmoprab 6093
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 6060 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
21dmeqi 5011 . 2  |-  dom  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  dom  {
<. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
3 dmopab 5020 . 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 1751 . . . . 5  |-  ( E. z E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. z ( w  =  <. x ,  y >.  /\  ph ) )
5 19.42v 1917 . . . . . 6  |-  ( E. z ( w  = 
<. x ,  y >.  /\  ph )  <->  ( w  =  <. x ,  y
>.  /\  E. z ph ) )
652exbii 1590 . . . . 5  |-  ( E. x E. y E. z ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  E. z ph )
)
74, 6bitri 241 . . . 4  |-  ( E. z E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  E. z ph )
)
87abbii 2499 . . 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 4208 . . 3  |-  { <. x ,  y >.  |  E. z ph }  =  {
w  |  E. x E. y ( w  = 
<. x ,  y >.  /\  E. z ph ) }
108, 9eqtr4i 2410 . 2  |-  { w  |  E. z E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) }  =  { <. x ,  y
>.  |  E. z ph }
112, 3, 103eqtri 2411 1  |-  dom  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  { <. x ,  y >.  |  E. z ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649   {cab 2373   <.cop 3760   {copab 4206   dom cdm 4818   {coprab 6021
This theorem is referenced by:  dmoprabss  6094  reldmoprab  6097  fnoprabg  6110  1st2val  6311  2nd2val  6312  linedegen  25791
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 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-dm 4828  df-oprab 6024
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