Theorem eldmcnv 38725

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 5846	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝐴◡𝑅𝑢))
2		brcnvg 5823	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ V) → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴))
3	2	elvd 3439	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴))
4	3	exbidv 1929	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑢 𝐴◡𝑅𝑢 ↔ ∃𝑢 𝑢𝑅𝐴))
5	1, 4	bitrd 281	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1787   ∈ wcel 2121  Vcvv 3433   class class class wbr 5074  ^◡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 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-cnv 5628  df-dm 5630

This theorem is referenced by:  eldmcoss  38928