Theorem eldmcnv 34654

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 5551	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝐴◡𝑅𝑢))
2		brcnvg 5534	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ V) → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴))
3	2	elvd 3419	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴))
4	3	exbidv 2020	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑢 𝐴◡𝑅𝑢 ↔ ∃𝑢 𝑢𝑅𝐴))
5	1, 4	bitrd 271	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

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