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Theorem eldmcnv 34654
 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 5551 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝐴𝑅𝑢))
2 brcnvg 5534 . . . 4 ((𝐴𝑉𝑢 ∈ V) → (𝐴𝑅𝑢𝑢𝑅𝐴))
32elvd 3419 . . 3 (𝐴𝑉 → (𝐴𝑅𝑢𝑢𝑅𝐴))
43exbidv 2020 . 2 (𝐴𝑉 → (∃𝑢 𝐴𝑅𝑢 ↔ ∃𝑢 𝑢𝑅𝐴))
51, 4bitrd 271 1 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∃wex 1878   ∈ wcel 2164  Vcvv 3414   class class class wbr 4873  ◡ccnv 5341  dom cdm 5342 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-cnv 5350  df-dm 5352 This theorem is referenced by:  eldmcoss  34749
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