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Theorem rnoprab2 5931
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 5930 . 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 2581 . . 3  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
32abbii 2395 . 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 2306 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 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   ran crn 4690   {coprab 5859
This theorem is referenced by:  rnmpt2  5954  prismorcsetlem  25912  prismorcset  25914  prismorcsetlemc  25917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700  df-oprab 5862
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