Theorem elinlem

Description: Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020)

	Ref	Expression
Assertion	elinlem	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ( I ‘ 𝐴 ) ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		elin	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) )
2		fvi	⊢ ( 𝐴 ∈ 𝐵 → ( I ‘ 𝐴 ) = 𝐴 )
3	2	eqcomd	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 = ( I ‘ 𝐴 ) )
4	3	eleq1d	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ 𝐶 ↔ ( I ‘ 𝐴 ) ∈ 𝐶 ) )
5	4	pm5.32i	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ( I ‘ 𝐴 ) ∈ 𝐶 ) )
6	1 5	bitri	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ( I ‘ 𝐴 ) ∈ 𝐶 ) )