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Theorem dfrn3 4822
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 4821 . 2  |-  ran  A  =  { y  |  E. x  x A y }
2 df-br 3964 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32exbii 1580 . . 3  |-  ( E. x  x A y  <->  E. x <. x ,  y
>.  e.  A )
43abbii 2368 . 2  |-  { y  |  E. x  x A y }  =  { y  |  E. x <. x ,  y
>.  e.  A }
51, 4eqtri 2276 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 2242   <.cop 3584   class class class wbr 3963   ran crn 4627
This theorem is referenced by:  elrn2g  4823  elrn2  4871  csbrng  4876  imadmrn  4977  imassrn  4978  prjrn  24414  csbrngVD  27685
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-cnv 4642  df-dm 4644  df-rn 4645
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