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Metamath Proof Explorer

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 ) )