Theorem eldm1cossres

Description: Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018)

	Ref	Expression
Assertion	eldm1cossres	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ dom ≀ ( 𝑅 ↾ 𝐴 ) ↔ ∃ 𝑢 ∈ 𝐴 𝑢 𝑅 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eldmcoss	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ dom ≀ ( 𝑅 ↾ 𝐴 ) ↔ ∃ 𝑢 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ) )
2		brres	⊢ ( 𝐵 ∈ 𝑉 → ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ↔ ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑅 𝐵 ) ) )
3	2	exbidv	⊢ ( 𝐵 ∈ 𝑉 → ( ∃ 𝑢 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ↔ ∃ 𝑢 ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑅 𝐵 ) ) )
4	1 3	bitrd	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ dom ≀ ( 𝑅 ↾ 𝐴 ) ↔ ∃ 𝑢 ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑅 𝐵 ) ) )
5		df-rex	⊢ ( ∃ 𝑢 ∈ 𝐴 𝑢 𝑅 𝐵 ↔ ∃ 𝑢 ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑅 𝐵 ) )
6	4 5	bitr4di	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ dom ≀ ( 𝑅 ↾ 𝐴 ) ↔ ∃ 𝑢 ∈ 𝐴 𝑢 𝑅 𝐵 ) )