Theorem eldmcnv 39080

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 5886	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝐴◡𝑅𝑢))
2		brcnvg 5863	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ V) → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴))
3	2	elvd 3459	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴))
4	3	exbidv 1954	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑢 𝐴◡𝑅𝑢 ↔ ∃𝑢 𝑢𝑅𝐴))
5	1, 4	bitrd 282	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812   ∈ wcel 2145  Vcvv 3453   class class class wbr 5107  ^◡ccnv 5658  dom cdm 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-cnv 5667  df-dm 5669

This theorem is used by:  eldmcoss  39283