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Theorem rnoprab 5747
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 5712 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
21rneqi 4678 . 2  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  ran  {
<. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
3 rnopab 4697 . 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 1626 . . . 4  |-  ( E. w E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y E. w ( w  =  <. x ,  y >.  /\  ph ) )
5 vex 2625 . . . . . . . 8  |-  x  e. 
_V
6 vex 2625 . . . . . . . 8  |-  y  e. 
_V
75, 6opex 4067 . . . . . . 7  |-  <. x ,  y >.  e.  _V
87isseti 2630 . . . . . 6  |-  E. w  w  =  <. x ,  y >.
9 19.41v 1831 . . . . . 6  |-  ( E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  ( E. w  w  =  <. x ,  y >.  /\  ph ) )
108, 9mpbiran 887 . . . . 5  |-  ( E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  ph )
11102exbii 1543 . . . 4  |-  ( E. x E. y E. w ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
124, 11bitri 183 . . 3  |-  ( E. w E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. x E. y ph )
1312abbii 2204 . 2  |-  { z  |  E. w E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  =  { z  |  E. x E. y ph }
142, 3, 133eqtri 2113 1  |-  ran  { <. <. x ,  y
>. ,  z >.  | 
ph }  =  {
z  |  E. x E. y ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1290   E.wex 1427   {cab 2075   <.cop 3455   {copab 3906   ran crn 4455   {coprab 5669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-cnv 4462  df-dm 4464  df-rn 4465  df-oprab 5672
This theorem is referenced by:  rnoprab2  5748
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