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Theorem rnoprab2 6149
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 6148 . 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 2735 . . 3  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
32abbii 2547 . 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 2458 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 1550    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   ran crn 4871   {coprab 6074
This theorem is referenced by:  rnmpt2  6172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-cnv 4878  df-dm 4880  df-rn 4881  df-oprab 6077
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