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Theorem dfrn2 4551
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 4384 . 2  |-  ran  A  =  dom  `' A
2 df-dm 4383 . 2  |-  dom  `' A  =  { y  |  E. x  y `' A x }
3 vex 2577 . . . . 5  |-  y  e. 
_V
4 vex 2577 . . . . 5  |-  x  e. 
_V
53, 4brcnv 4546 . . . 4  |-  ( y `' A x  <->  x A
y )
65exbii 1512 . . 3  |-  ( E. x  y `' A x 
<->  E. x  x A y )
76abbii 2169 . 2  |-  { y  |  E. x  y `' A x }  =  { y  |  E. x  x A y }
81, 2, 73eqtri 2080 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:    = wceq 1259   E.wex 1397   {cab 2042   class class class wbr 3792   `'ccnv 4372   dom cdm 4373   ran crn 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-cnv 4381  df-dm 4383  df-rn 4384
This theorem is referenced by:  dfrn3  4552  dfdm4  4555  dm0rn0  4580  dmmrnm  4582  dfrnf  4603  dfima2  4698  funcnv3  4989
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