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Theorem releldmb 5939
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 5892 . . 3 (𝐴 ∈ dom 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
21ibi 267 . 2 (𝐴 ∈ dom 𝑅 → ∃𝑥 𝐴𝑅𝑥)
3 releldm 5937 . . . 4 ((Rel 𝑅𝐴𝑅𝑥) → 𝐴 ∈ dom 𝑅)
43ex 412 . . 3 (Rel 𝑅 → (𝐴𝑅𝑥𝐴 ∈ dom 𝑅))
54exlimdv 1928 . 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 1773  wcel 2098   class class class wbr 5141  dom cdm 5669  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-dm 5679
This theorem is referenced by:  eqvrelref  37993
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