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Theorem dfrn3 4698
Description: Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
Assertion
Ref Expression
dfrn3  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
Distinct variable group:    x, y, A

Proof of Theorem dfrn3
StepHypRef Expression
1 dfrn2 4697 . 2  |-  ran  A  =  { y  |  E. x  x A y }
2 df-br 3900 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32exbii 1569 . . 3  |-  ( E. x  x A y  <->  E. x <. x ,  y
>.  e.  A )
43abbii 2233 . 2  |-  { y  |  E. x  x A y }  =  { y  |  E. x <. x ,  y
>.  e.  A }
51, 4eqtri 2138 1  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   <.cop 3500   class class class wbr 3899   ran crn 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  elrn2g  4699  elrn2  4751  imadmrn  4861  imassrn  4862  csbrng  4970
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