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Theorem releldmb 5945
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 5898 . . 3 (𝐴 ∈ dom 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
21ibi 267 . 2 (𝐴 ∈ dom 𝑅 → ∃𝑥 𝐴𝑅𝑥)
3 releldm 5943 . . . 4 ((Rel 𝑅𝐴𝑅𝑥) → 𝐴 ∈ dom 𝑅)
43ex 412 . . 3 (Rel 𝑅 → (𝐴𝑅𝑥𝐴 ∈ dom 𝑅))
54exlimdv 1935 . 2 (Rel 𝑅 → (∃𝑥 𝐴𝑅𝑥𝐴 ∈ dom 𝑅))
62, 5impbid2 225 1 (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1780  wcel 2105   class class class wbr 5148  dom cdm 5676  Rel wrel 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-dm 5686
This theorem is referenced by:  eqvrelref  37943
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