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Theorem dfrn2 4523
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 4356 . 2  |-  ran  A  =  dom  `' A
2 df-dm 4355 . 2  |-  dom  `' A  =  { y  |  E. x  y `' A x }
3 vex 2560 . . . . 5  |-  y  e. 
_V
4 vex 2560 . . . . 5  |-  x  e. 
_V
53, 4brcnv 4518 . . . 4  |-  ( y `' A x  <->  x A
y )
65exbii 1496 . . 3  |-  ( E. x  y `' A x 
<->  E. x  x A y )
76abbii 2153 . 2  |-  { y  |  E. x  y `' A x }  =  { y  |  E. x  x A y }
81, 2, 73eqtri 2064 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:    = wceq 1243   E.wex 1381   {cab 2026   class class class wbr 3764   `'ccnv 4344   dom cdm 4345   ran crn 4346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356
This theorem is referenced by:  dfrn3  4524  dfdm4  4527  dm0rn0  4552  dmmrnm  4554  dfrnf  4575  dfima2  4670  funcnv3  4961
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