Theorem eldmcoss2

Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018)

	Ref	Expression
Assertion	eldmcoss2	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eldmcoss	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ dom ≀ 𝑅 ↔ ∃ 𝑢 𝑢 𝑅 𝐴 ) )
2		brcoss	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ≀ 𝑅 𝐴 ↔ ∃ 𝑢 ( 𝑢 𝑅 𝐴 ∧ 𝑢 𝑅 𝐴 ) ) )
3	2	anidms	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ≀ 𝑅 𝐴 ↔ ∃ 𝑢 ( 𝑢 𝑅 𝐴 ∧ 𝑢 𝑅 𝐴 ) ) )
4		pm4.24	⊢ ( 𝑢 𝑅 𝐴 ↔ ( 𝑢 𝑅 𝐴 ∧ 𝑢 𝑅 𝐴 ) )
5	4	exbii	⊢ ( ∃ 𝑢 𝑢 𝑅 𝐴 ↔ ∃ 𝑢 ( 𝑢 𝑅 𝐴 ∧ 𝑢 𝑅 𝐴 ) )
6	3 5	bitr4di	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ≀ 𝑅 𝐴 ↔ ∃ 𝑢 𝑢 𝑅 𝐴 ) )
7	1 6	bitr4d	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅 𝐴 ) )