Theorem eldmcnv 38327

Description: Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.)

Assertion

Ref	Expression
eldmcnv	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑉

Proof of Theorem eldmcnv

Step	Hyp	Ref	Expression
1		eldmg 5912	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝐴◡𝑅𝑢))
2		brcnvg 5893	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ V) → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴))
3	2	elvd 3484	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴))
4	3	exbidv 1919	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑢 𝐴◡𝑅𝑢 ↔ ∃𝑢 𝑢𝑅𝐴))
5	1, 4	bitrd 279	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1776   ∈ wcel 2106  Vcvv 3478   class class class wbr 5148  ^◡ccnv 5688  dom cdm 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699

This theorem is referenced by:  eldmcoss  38440