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Theorem dfrn2 4856
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 4680 . 2  |-  ran  A  =  dom  `'  A
2 df-dm 4679 . 2  |-  dom  `'  A  =  { y  |  E. x  y `' A x }
3 vex 2766 . . . . 5  |-  y  e. 
_V
4 vex 2766 . . . . 5  |-  x  e. 
_V
53, 4brcnv 4852 . . . 4  |-  ( y `' A x  <->  x A
y )
65exbii 1580 . . 3  |-  ( E. x  y `' A x 
<->  E. x  x A y )
76abbii 2370 . 2  |-  { y  |  E. x  y `' A x }  =  { y  |  E. x  x A y }
81, 2, 73eqtri 2282 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:   E.wex 1537    = wceq 1619   {cab 2244   class class class wbr 3997   `'ccnv 4660   dom cdm 4661   ran crn 4662
This theorem is referenced by:  dfrn3  4857  dfdm4  4860  dm0rn0  4883  dfrnf  4905  dfima2  5002  funcnv3  5249
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-cnv 4677  df-dm 4679  df-rn 4680
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