MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfrn3 Structured version   Visualization version   GIF version

Theorem dfrn3 5222
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 5221 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
2 df-br 4578 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32exbii 1763 . . 3 (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
43abbii 2725 . 2 {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
51, 4eqtri 2631 1 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wex 1694  wcel 1976  {cab 2595  cop 4130   class class class wbr 4577  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:  elrn2g  5223  elrn2  5273  imadmrn  5382  imassrn  5383  csbrngOLD  37874  csbrngVD  37950
  Copyright terms: Public domain W3C validator