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Theorem elrng 5902
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 5901 . 2 (𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))
2 df-br 5144 . . 3 (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
32exbii 1848 . 2 (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
41, 3bitr4di 289 1 (𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wex 1779  wcel 2108  cop 4632   class class class wbr 5143  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  elrn  5904  ssrelrn  5905  relelrnb  5958  cicsym  17848  elrnres  38272
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