Metamath Proof Explorer


Theorem intimasn

Description: Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020)

Ref Expression
Assertion intimasn ( 𝐵𝑉 → ( 𝐴 “ { 𝐵 } ) = { 𝑥 ∣ ∃ 𝑎𝐴 𝑥 = ( 𝑎 “ { 𝐵 } ) } )

Proof

Step Hyp Ref Expression
1 ax-5 ( 𝐵𝑉 → ∀ 𝑦 𝐵𝑉 )
2 r19.12sn ( 𝐵𝑉 → ( ∃ 𝑏 ∈ { 𝐵 } ∀ 𝑎𝐴𝑏 , 𝑦 ⟩ ∈ 𝑎 ↔ ∀ 𝑎𝐴𝑏 ∈ { 𝐵 } ⟨ 𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
3 2 biimprd ( 𝐵𝑉 → ( ∀ 𝑎𝐴𝑏 ∈ { 𝐵 } ⟨ 𝑏 , 𝑦 ⟩ ∈ 𝑎 → ∃ 𝑏 ∈ { 𝐵 } ∀ 𝑎𝐴𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
4 3 alimi ( ∀ 𝑦 𝐵𝑉 → ∀ 𝑦 ( ∀ 𝑎𝐴𝑏 ∈ { 𝐵 } ⟨ 𝑏 , 𝑦 ⟩ ∈ 𝑎 → ∃ 𝑏 ∈ { 𝐵 } ∀ 𝑎𝐴𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
5 intimag ( ∀ 𝑦 ( ∀ 𝑎𝐴𝑏 ∈ { 𝐵 } ⟨ 𝑏 , 𝑦 ⟩ ∈ 𝑎 → ∃ 𝑏 ∈ { 𝐵 } ∀ 𝑎𝐴𝑏 , 𝑦 ⟩ ∈ 𝑎 ) → ( 𝐴 “ { 𝐵 } ) = { 𝑥 ∣ ∃ 𝑎𝐴 𝑥 = ( 𝑎 “ { 𝐵 } ) } )
6 1 4 5 3syl ( 𝐵𝑉 → ( 𝐴 “ { 𝐵 } ) = { 𝑥 ∣ ∃ 𝑎𝐴 𝑥 = ( 𝑎 “ { 𝐵 } ) } )