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Theorem dfrn2 4867
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 4699 . 2  |-  ran  A  =  dom  `'  A
2 df-dm 4698 . 2  |-  dom  `'  A  =  { y  |  E. x  y `' A x }
3 vex 2792 . . . . 5  |-  y  e. 
_V
4 vex 2792 . . . . 5  |-  x  e. 
_V
53, 4brcnv 4863 . . . 4  |-  ( y `' A x  <->  x A
y )
65exbii 1570 . . 3  |-  ( E. x  y `' A x 
<->  E. x  x A y )
76abbii 2396 . 2  |-  { y  |  E. x  y `' A x }  =  { y  |  E. x  x A y }
81, 2, 73eqtri 2308 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:   E.wex 1529    = wceq 1624   {cab 2270   class class class wbr 4024   `'ccnv 4687   dom cdm 4688   ran crn 4689
This theorem is referenced by:  dfrn3  4868  dfdm4  4871  dm0rn0  4894  dfrnf  4916  dfima2  5013  funcnv3  5276
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-cnv 4696  df-dm 4698  df-rn 4699
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