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Theorem dfrn3 4869
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 4868 . 2  |-  ran  A  =  { y  |  E. x  x A y }
2 df-br 4026 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32exbii 1570 . . 3  |-  ( E. x  x A y  <->  E. x <. x ,  y
>.  e.  A )
43abbii 2397 . 2  |-  { y  |  E. x  x A y }  =  { y  |  E. x <. x ,  y
>.  e.  A }
51, 4eqtri 2305 1  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
Colors of variables: wff set class
Syntax hints:   E.wex 1529    = wceq 1624    e. wcel 1685   {cab 2271   <.cop 3645   class class class wbr 4025   ran crn 4690
This theorem is referenced by:  elrn2g  4870  elrn2  4918  csbrng  4923  imadmrn  5024  imassrn  5025  prjrn  24482  csbrngVD  27941
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-cnv 4697  df-dm 4699  df-rn 4700
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