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Theorem dfrn3 5449
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 5448 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
2 df-br 4788 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32exbii 1924 . . 3 (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
43abbii 2888 . 2 {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
51, 4eqtri 2793 1 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wex 1852  wcel 2145  {cab 2757  cop 4323   class class class wbr 4787  ran crn 5251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-cnv 5258  df-dm 5260  df-rn 5261
This theorem is referenced by:  elrn2g  5450  elrn2  5502  imadmrn  5616  imassrn  5617  csbrngOLD  39579  csbrngVD  39654
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