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Theorem dfrn3 5725
 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 5724 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
2 df-br 5032 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32exbii 1849 . . 3 (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
43abbii 2863 . 2 {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
51, 4eqtri 2821 1 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ⟨cop 4531   class class class wbr 5031  ran crn 5521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-cnv 5528  df-dm 5530  df-rn 5531 This theorem is referenced by:  elrn2g  5726  elrn2  5786  imadmrn  5907  imassrn  5908  csbrngVD  41645
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