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Theorem rnoprab2 5947
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 5946 . 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 2594 . . 3  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
32abbii 2408 . 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 2319 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   ran crn 4706   {coprab 5875
This theorem is referenced by:  rnmpt2  5970  prismorcsetlem  26015  prismorcset  26017  prismorcsetlemc  26020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-oprab 5878
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