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Theorem rnoprab 6158
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 6123 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
21rneqi 5098 . 2  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  ran  {
<. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
3 rnopab 5117 . 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 1760 . . . 4  |-  ( E. w E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. w ( w  =  <. x ,  y >.  /\  ph ) )
5 opex 4429 . . . . . . 7  |-  <. x ,  y >.  e.  _V
65isseti 2964 . . . . . 6  |-  E. w  w  =  <. x ,  y >.
7 19.41v 1925 . . . . . 6  |-  ( E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  ( E. w  w  =  <. x ,  y >.  /\  ph ) )
86, 7mpbiran 886 . . . . 5  |-  ( E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  ph )
982exbii 1594 . . . 4  |-  ( E. x E. y E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
104, 9bitri 242 . . 3  |-  ( E. w E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
1110abbii 2550 . 2  |-  { z  |  E. w E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  =  { z  |  E. x E. y ph }
122, 3, 113eqtri 2462 1  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  {
z  |  E. x E. y ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1551    = wceq 1653   {cab 2424   <.cop 3819   {copab 4267   ran crn 4881   {coprab 6084
This theorem is referenced by:  rnoprab2  6159  ellines  26088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-cnv 4888  df-dm 4890  df-rn 4891  df-oprab 6087
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