Theorem elfi

Description: Specific properties of an element of ( fiB ) . (Contributed by FL, 27-Apr-2008) (Revised by Mario Carneiro, 24-Nov-2013)

	Ref	Expression
Assertion	elfi	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∈ ( fi ‘ 𝐵 ) ↔ ∃ 𝑥 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝐴 = ∩ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		fival	⊢ ( 𝐵 ∈ 𝑊 → ( fi ‘ 𝐵 ) = { 𝑦 ∣ ∃ 𝑥 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑦 = ∩ 𝑥 } )
2	1	eleq2d	⊢ ( 𝐵 ∈ 𝑊 → ( 𝐴 ∈ ( fi ‘ 𝐵 ) ↔ 𝐴 ∈ { 𝑦 ∣ ∃ 𝑥 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑦 = ∩ 𝑥 } ) )
3		eqeq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 = ∩ 𝑥 ↔ 𝐴 = ∩ 𝑥 ) )
4	3	rexbidv	⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑦 = ∩ 𝑥 ↔ ∃ 𝑥 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝐴 = ∩ 𝑥 ) )
5	4	elabg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ { 𝑦 ∣ ∃ 𝑥 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑦 = ∩ 𝑥 } ↔ ∃ 𝑥 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝐴 = ∩ 𝑥 ) )
6	2 5	sylan9bbr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∈ ( fi ‘ 𝐵 ) ↔ ∃ 𝑥 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝐴 = ∩ 𝑥 ) )