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Theorem eldmcnv 36619
Description: Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.)
Assertion
Ref Expression
eldmcnv (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑉

Proof of Theorem eldmcnv
StepHypRef Expression
1 eldmg 5840 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝐴𝑅𝑢))
2 brcnvg 5821 . . . 4 ((𝐴𝑉𝑢 ∈ V) → (𝐴𝑅𝑢𝑢𝑅𝐴))
32elvd 3448 . . 3 (𝐴𝑉 → (𝐴𝑅𝑢𝑢𝑅𝐴))
43exbidv 1923 . 2 (𝐴𝑉 → (∃𝑢 𝐴𝑅𝑢 ↔ ∃𝑢 𝑢𝑅𝐴))
51, 4bitrd 278 1 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1780  wcel 2105  Vcvv 3441   class class class wbr 5092  ccnv 5619  dom cdm 5620
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 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-cnv 5628  df-dm 5630
This theorem is referenced by:  eldmcoss  36733
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