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Theorem dfrn2 5221
Description: Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
Assertion
Ref Expression
dfrn2 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfrn2
StepHypRef Expression
1 df-rn 5039 . 2 ran 𝐴 = dom 𝐴
2 df-dm 5038 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
3 vex 3175 . . . . 5 𝑦 ∈ V
4 vex 3175 . . . . 5 𝑥 ∈ V
53, 4brcnv 5215 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
65exbii 1763 . . 3 (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
76abbii 2725 . 2 {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
81, 2, 73eqtri 2635 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wex 1694  {cab 2595   class class class wbr 4577  ccnv 5027  dom cdm 5028  ran crn 5029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-cnv 5036  df-dm 5038  df-rn 5039
This theorem is referenced by:  dfrn3  5222  dfdm4  5225  dm0rn0  5250  dfrnf  5272  dfima2  5374  funcnv3  5859  opabrn  28610
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