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Theorem dfrn3 5753
Description: Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
Assertion
Ref Expression
dfrn3 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfrn3
StepHypRef Expression
1 dfrn2 5752 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
2 df-br 5058 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32exbii 1839 . . 3 (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
43abbii 2883 . 2 {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
51, 4eqtri 2841 1 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wex 1771  wcel 2105  {cab 2796  cop 4563   class class class wbr 5057  ran crn 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  elrn2g  5754  elrn2  5814  imadmrn  5932  imassrn  5933  csbrngVD  41107
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