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Theorem eldmcnv 34448
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 5455 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝐴𝑅𝑢))
2 brcnvg 5439 . . . 4 ((𝐴𝑉𝑢 ∈ V) → (𝐴𝑅𝑢𝑢𝑅𝐴))
32el2v2 34324 . . 3 (𝐴𝑉 → (𝐴𝑅𝑢𝑢𝑅𝐴))
43exbidv 2002 . 2 (𝐴𝑉 → (∃𝑢 𝐴𝑅𝑢 ↔ ∃𝑢 𝑢𝑅𝐴))
51, 4bitrd 268 1 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wex 1852  wcel 2145  Vcvv 3351   class class class wbr 4786  ccnv 5248  dom cdm 5249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-cnv 5257  df-dm 5259
This theorem is referenced by:  eldmcoss  34543
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