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Theorem dmoprab 6146
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 6113 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
21dmeqi 5063 . 2  |-  dom  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  dom  {
<. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
3 dmopab 5072 . 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 1759 . . . . 5  |-  ( E. z E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. z ( w  =  <. x ,  y >.  /\  ph ) )
5 19.42v 1928 . . . . . 6  |-  ( E. z ( w  = 
<. x ,  y >.  /\  ph )  <->  ( w  =  <. x ,  y
>.  /\  E. z ph ) )
652exbii 1593 . . . . 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 2547 . . 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 4259 . . 3  |-  { <. x ,  y >.  |  E. z ph }  =  {
w  |  E. x E. y ( w  = 
<. x ,  y >.  /\  E. z ph ) }
108, 9eqtr4i 2458 . 2  |-  { w  |  E. z E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) }  =  { <. x ,  y
>.  |  E. z ph }
112, 3, 103eqtri 2459 1  |-  dom  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  { <. x ,  y >.  |  E. z ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652   {cab 2421   <.cop 3809   {copab 4257   dom cdm 4870   {coprab 6074
This theorem is referenced by:  dmoprabss  6147  reldmoprab  6150  fnoprabg  6163  1st2val  6364  2nd2val  6365  linedegen  26069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-dm 4880  df-oprab 6077
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