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Theorem rnoprab 6366
Description: The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
Assertion
Ref Expression
rnoprab  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  {
z  |  E. x E. y ph }
Distinct variable groups:    x, z    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem rnoprab
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dfoprab2 6324 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
21rneqi 5215 . 2  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  ran  {
<. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
3 rnopab 5233 . 2  |-  ran  { <. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
z  |  E. w E. x E. y ( w  =  <. x ,  y >.  /\  ph ) }
4 exrot3 1836 . . . 4  |-  ( E. w E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. w ( w  =  <. x ,  y >.  /\  ph ) )
5 opex 4697 . . . . . . 7  |-  <. x ,  y >.  e.  _V
65isseti 3099 . . . . . 6  |-  E. w  w  =  <. x ,  y >.
7 19.41v 1755 . . . . . 6  |-  ( E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  ( E. w  w  =  <. x ,  y >.  /\  ph ) )
86, 7mpbiran 916 . . . . 5  |-  ( E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  ph )
982exbii 1653 . . . 4  |-  ( E. x E. y E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
104, 9bitri 249 . . 3  |-  ( E. w E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
1110abbii 2575 . 2  |-  { z  |  E. w E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  =  { z  |  E. x E. y ph }
122, 3, 113eqtri 2474 1  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  {
z  |  E. x E. y ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1381   E.wex 1597   {cab 2426   <.cop 4016   {copab 4490   ran crn 4986   {coprab 6278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-cnv 4993  df-dm 4995  df-rn 4996  df-oprab 6281
This theorem is referenced by:  rnoprab2  6367  elrnmpt2res  6397  ellines  29770
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