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Theorem rnoprab 6284
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 6242 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
21rneqi 5142 . 2  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  ran  {
<. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
3 rnopab 5160 . 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 1860 . . . 4  |-  ( E. w E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. w ( w  =  <. x ,  y >.  /\  ph ) )
5 opex 4626 . . . . . . 7  |-  <. x ,  y >.  e.  _V
65isseti 3040 . . . . . 6  |-  E. w  w  =  <. x ,  y >.
7 19.41v 1779 . . . . . 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 1676 . . . 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 2516 . 2  |-  { z  |  E. w E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  =  { z  |  E. x E. y ph }
122, 3, 113eqtri 2415 1  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  {
z  |  E. x E. y ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1399   E.wex 1620   {cab 2367   <.cop 3950   {copab 4424   ran crn 4914   {coprab 6197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-cnv 4921  df-dm 4923  df-rn 4924  df-oprab 6200
This theorem is referenced by:  rnoprab2  6285  elrnmpt2res  6315  ellines  29955
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