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Theorem rnoprab2 6096
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 6095 . 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 2687 . . 3  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
32abbii 2499 . 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 2410 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 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2373   E.wrex 2650   ran crn 4819   {coprab 6021
This theorem is referenced by:  rnmpt2  6119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-cnv 4826  df-dm 4828  df-rn 4829  df-oprab 6024
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