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Theorem dfrn2 4775
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 4599 . 2  |-  ran  A  =  dom  `'  A
2 df-dm 4598 . 2  |-  dom  `'  A  =  { y  |  E. x  y `' A x }
3 vex 2730 . . . . 5  |-  y  e. 
_V
4 vex 2730 . . . . 5  |-  x  e. 
_V
53, 4brcnv 4771 . . . 4  |-  ( y `' A x  <->  x A
y )
65exbii 1580 . . 3  |-  ( E. x  y `' A x 
<->  E. x  x A y )
76abbii 2361 . 2  |-  { y  |  E. x  y `' A x }  =  { y  |  E. x  x A y }
81, 2, 73eqtri 2277 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:   E.wex 1537    = wceq 1619   {cab 2239   class class class wbr 3920   `'ccnv 4579   dom cdm 4580   ran crn 4581
This theorem is referenced by:  dfrn3  4776  dfdm4  4779  dm0rn0  4802  dfrnf  4824  dfima2  4921  funcnv3  5168
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-cnv 4596  df-dm 4598  df-rn 4599
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