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Theorem dfrn2 4868
Description: Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
Assertion
Ref Expression
dfrn2  |-  ran  A  =  { y  |  E. x  x A y }
Distinct variable group:    x, y, A

Proof of Theorem dfrn2
StepHypRef Expression
1 df-rn 4700 . 2  |-  ran  A  =  dom  `' A
2 df-dm 4699 . 2  |-  dom  `' A  =  { y  |  E. x  y `' A x }
3 vex 2791 . . . . 5  |-  y  e. 
_V
4 vex 2791 . . . . 5  |-  x  e. 
_V
53, 4brcnv 4864 . . . 4  |-  ( y `' A x  <->  x A
y )
65exbii 1569 . . 3  |-  ( E. x  y `' A x 
<->  E. x  x A y )
76abbii 2395 . 2  |-  { y  |  E. x  y `' A x }  =  { y  |  E. x  x A y }
81, 2, 73eqtri 2307 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623   {cab 2269   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690
This theorem is referenced by:  dfrn3  4869  dfdm4  4872  dm0rn0  4895  dfrnf  4917  dfima2  5014  funcnv3  5311
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700
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