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Theorem dfrn2 5891
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 5689 . 2 ran 𝐴 = dom 𝐴
2 df-dm 5688 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
3 vex 3475 . . . . 5 𝑦 ∈ V
4 vex 3475 . . . . 5 𝑥 ∈ V
53, 4brcnv 5885 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
65exbii 1843 . . 3 (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
76abbii 2798 . 2 {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
81, 2, 73eqtri 2760 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wex 1774  {cab 2705   class class class wbr 5148  ccnv 5677  dom cdm 5678  ran crn 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-cnv 5686  df-dm 5688  df-rn 5689
This theorem is referenced by:  dfrn3  5892  dfdm4  5898  dm0rn0  5927  rnep  5929  dfrnf  5952  dfima2  6065  funcnv3  6623  opabrn  32415  rncossdmcoss  37927
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