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Theorem dfrn2 5059
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 4889 . 2  |-  ran  A  =  dom  `' A
2 df-dm 4888 . 2  |-  dom  `' A  =  { y  |  E. x  y `' A x }
3 vex 2959 . . . . 5  |-  y  e. 
_V
4 vex 2959 . . . . 5  |-  x  e. 
_V
53, 4brcnv 5055 . . . 4  |-  ( y `' A x  <->  x A
y )
65exbii 1592 . . 3  |-  ( E. x  y `' A x 
<->  E. x  x A y )
76abbii 2548 . 2  |-  { y  |  E. x  y `' A x }  =  { y  |  E. x  x A y }
81, 2, 73eqtri 2460 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:   E.wex 1550    = wceq 1652   {cab 2422   class class class wbr 4212   `'ccnv 4877   dom cdm 4878   ran crn 4879
This theorem is referenced by:  dfrn3  5060  dfdm4  5063  dm0rn0  5086  dfrnf  5108  dfima2  5205  funcnv3  5512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cnv 4886  df-dm 4888  df-rn 4889
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