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Theorem dfrn2 4612
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 4439 . 2  |-  ran  A  =  dom  `' A
2 df-dm 4438 . 2  |-  dom  `' A  =  { y  |  E. x  y `' A x }
3 vex 2622 . . . . 5  |-  y  e. 
_V
4 vex 2622 . . . . 5  |-  x  e. 
_V
53, 4brcnv 4607 . . . 4  |-  ( y `' A x  <->  x A
y )
65exbii 1541 . . 3  |-  ( E. x  y `' A x 
<->  E. x  x A y )
76abbii 2203 . 2  |-  { y  |  E. x  y `' A x }  =  { y  |  E. x  x A y }
81, 2, 73eqtri 2112 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:    = wceq 1289   E.wex 1426   {cab 2074   class class class wbr 3837   `'ccnv 4427   dom cdm 4428   ran crn 4429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  dfrn3  4613  dfdm4  4616  dm0rn0  4641  dmmrnm  4643  dfrnf  4664  dfima2  4763  funcnv3  5062
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