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Theorem releldmb 5960
Description: Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
Assertion
Ref Expression
releldmb (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem releldmb
StepHypRef Expression
1 eldmg 5912 . . 3 (𝐴 ∈ dom 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
21ibi 267 . 2 (𝐴 ∈ dom 𝑅 → ∃𝑥 𝐴𝑅𝑥)
3 releldm 5958 . . . 4 ((Rel 𝑅𝐴𝑅𝑥) → 𝐴 ∈ dom 𝑅)
43ex 412 . . 3 (Rel 𝑅 → (𝐴𝑅𝑥𝐴 ∈ dom 𝑅))
54exlimdv 1931 . 2 (Rel 𝑅 → (∃𝑥 𝐴𝑅𝑥𝐴 ∈ dom 𝑅))
62, 5impbid2 226 1 (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wex 1776  wcel 2106   class class class wbr 5148  dom cdm 5689  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699
This theorem is referenced by:  eqvrelref  38592
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