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Theorem rnoprab2 5613
Description: The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)
Assertion
Ref Expression
rnoprab2  |-  ran  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  =  {
z  |  E. x  e.  A  E. y  e.  B  ph }
Distinct variable groups:    y, A    x, y, z
Allowed substitution hints:    ph( x, y, z)    A( x, z)    B( x, y, z)

Proof of Theorem rnoprab2
StepHypRef Expression
1 rnoprab 5612 . 2  |-  ran  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  =  {
z  |  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }
2 r2ex 2387 . . 3  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
32abbii 2195 . 2  |-  { z  |  E. x  e.  A  E. y  e.  B  ph }  =  { z  |  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }
41, 3eqtr4i 2105 1  |-  ran  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  =  {
z  |  E. x  e.  A  E. y  e.  B  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2068   E.wrex 2350   ran crn 4366   {coprab 5538
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-cnv 4373  df-dm 4375  df-rn 4376  df-oprab 5541
This theorem is referenced by:  rnmpt2  5636
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