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Theorem dfrn3 4788
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 4787 . 2  |-  ran  A  =  { y  |  E. x  x A y }
2 df-br 3978 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32exbii 1592 . . 3  |-  ( E. x  x A y  <->  E. x <. x ,  y
>.  e.  A )
43abbii 2280 . 2  |-  { y  |  E. x  x A y }  =  { y  |  E. x <. x ,  y
>.  e.  A }
51, 4eqtri 2185 1  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1342   E.wex 1479    e. wcel 2135   {cab 2150   <.cop 3574   class class class wbr 3977   ran crn 4600
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-cnv 4607  df-dm 4609  df-rn 4610
This theorem is referenced by:  elrn2g  4789  elrn2  4841  imadmrn  4951  imassrn  4952  csbrng  5060
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