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

Theorem elrng 5755
Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
elrng (𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elrng
StepHypRef Expression
1 elrn2g 5754 . 2 (𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))
2 df-br 5058 . . 3 (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
32exbii 1839 . 2 (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
41, 3syl6bbr 290 1 (𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wex 1771  wcel 2105  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:  ssrelrn  5756  relelrnb  5810  cicsym  17062  trpredpred  32964
  Copyright terms: Public domain W3C validator