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Theorem dfrn2 4774
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 4597 . 2  |-  ran  A  =  dom  `' A
2 df-dm 4596 . 2  |-  dom  `' A  =  { y  |  E. x  y `' A x }
3 vex 2715 . . . . 5  |-  y  e. 
_V
4 vex 2715 . . . . 5  |-  x  e. 
_V
53, 4brcnv 4769 . . . 4  |-  ( y `' A x  <->  x A
y )
65exbii 1585 . . 3  |-  ( E. x  y `' A x 
<->  E. x  x A y )
76abbii 2273 . 2  |-  { y  |  E. x  y `' A x }  =  { y  |  E. x  x A y }
81, 2, 73eqtri 2182 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:    = wceq 1335   E.wex 1472   {cab 2143   class class class wbr 3965   `'ccnv 4585   dom cdm 4586   ran crn 4587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594  df-dm 4596  df-rn 4597
This theorem is referenced by:  dfrn3  4775  dfdm4  4778  dm0rn0  4803  dmmrnm  4805  dfrnf  4827  dfima2  4930  funcnv3  5232
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