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Theorem dfrn2 4884
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 4716 . 2  |-  ran  A  =  dom  `' A
2 df-dm 4715 . 2  |-  dom  `' A  =  { y  |  E. x  y `' A x }
3 vex 2804 . . . . 5  |-  y  e. 
_V
4 vex 2804 . . . . 5  |-  x  e. 
_V
53, 4brcnv 4880 . . . 4  |-  ( y `' A x  <->  x A
y )
65exbii 1572 . . 3  |-  ( E. x  y `' A x 
<->  E. x  x A y )
76abbii 2408 . 2  |-  { y  |  E. x  y `' A x }  =  { y  |  E. x  x A y }
81, 2, 73eqtri 2320 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:   E.wex 1531    = wceq 1632   {cab 2282   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ran crn 4706
This theorem is referenced by:  dfrn3  4885  dfdm4  4888  dm0rn0  4911  dfrnf  4933  dfima2  5030  funcnv3  5327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716
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