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Theorem rnoprab 6406
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 6364 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
21rneqi 5080 . 2  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  ran  {
<. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
3 rnopab 5098 . 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 1942 . . . 4  |-  ( E. w E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. w ( w  =  <. x ,  y >.  /\  ph ) )
5 opex 4678 . . . . . . 7  |-  <. x ,  y >.  e.  _V
65isseti 3063 . . . . . 6  |-  E. w  w  =  <. x ,  y >.
7 19.41v 1841 . . . . . 6  |-  ( E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  ( E. w  w  =  <. x ,  y >.  /\  ph ) )
86, 7mpbiran 934 . . . . 5  |-  ( E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  ph )
982exbii 1730 . . . 4  |-  ( E. x E. y E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
104, 9bitri 257 . . 3  |-  ( E. w E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
1110abbii 2578 . 2  |-  { z  |  E. w E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  =  { z  |  E. x E. y ph }
122, 3, 113eqtri 2488 1  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  {
z  |  E. x E. y ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 375    = wceq 1455   E.wex 1674   {cab 2448   <.cop 3986   {copab 4474   ran crn 4854   {coprab 6316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-cnv 4861  df-dm 4863  df-rn 4864  df-oprab 6319
This theorem is referenced by:  rnoprab2  6407  elrnmpt2res  6437  ellines  30968
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