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Theorem elrng 5901
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 5900 . 2 (𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))
2 df-br 5143 . . 3 (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
32exbii 1847 . 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 1778  wcel 2107  cop 4631   class class class wbr 5142  ran crn 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-cnv 5692  df-dm 5694  df-rn 5695
This theorem is referenced by:  elrn  5903  ssrelrn  5904  relelrnb  5957  cicsym  17849  elrnres  38273
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