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Theorem dfrn3 4776
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 4775 . 2  |-  ran  A  =  { y  |  E. x  x A y }
2 df-br 3921 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32exbii 1580 . . 3  |-  ( E. x  x A y  <->  E. x <. x ,  y
>.  e.  A )
43abbii 2361 . 2  |-  { y  |  E. x  x A y }  =  { y  |  E. x <. x ,  y
>.  e.  A }
51, 4eqtri 2273 1  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
Colors of variables: wff set class
Syntax hints:   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2239   <.cop 3547   class class class wbr 3920   ran crn 4581
This theorem is referenced by:  elrn2g  4777  elrn2  4825  csbrng  4830  imadmrn  4931  imassrn  4932  prjrn  24248  csbrngVD  27459
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-cnv 4596  df-dm 4598  df-rn 4599
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