Metamath Proof Explorer


Theorem bj-restsn

Description: An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 and bj-restsnid . (Contributed by BJ, 27-Apr-2021)

Ref Expression
Assertion bj-restsn ( ( 𝑌𝑉𝐴𝑊 ) → ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌𝐴 ) } )

Proof

Step Hyp Ref Expression
1 snex { 𝑌 } ∈ V
2 elrest ( ( { 𝑌 } ∈ V ∧ 𝐴𝑊 ) → ( 𝑥 ∈ ( { 𝑌 } ↾t 𝐴 ) ↔ ∃ 𝑦 ∈ { 𝑌 } 𝑥 = ( 𝑦𝐴 ) ) )
3 1 2 mpan ( 𝐴𝑊 → ( 𝑥 ∈ ( { 𝑌 } ↾t 𝐴 ) ↔ ∃ 𝑦 ∈ { 𝑌 } 𝑥 = ( 𝑦𝐴 ) ) )
4 velsn ( 𝑥 ∈ { ( 𝑦𝐴 ) } ↔ 𝑥 = ( 𝑦𝐴 ) )
5 ineq1 ( 𝑦 = 𝑌 → ( 𝑦𝐴 ) = ( 𝑌𝐴 ) )
6 5 sneqd ( 𝑦 = 𝑌 → { ( 𝑦𝐴 ) } = { ( 𝑌𝐴 ) } )
7 6 eleq2d ( 𝑦 = 𝑌 → ( 𝑥 ∈ { ( 𝑦𝐴 ) } ↔ 𝑥 ∈ { ( 𝑌𝐴 ) } ) )
8 4 7 bitr3id ( 𝑦 = 𝑌 → ( 𝑥 = ( 𝑦𝐴 ) ↔ 𝑥 ∈ { ( 𝑌𝐴 ) } ) )
9 8 rexsng ( 𝑌𝑉 → ( ∃ 𝑦 ∈ { 𝑌 } 𝑥 = ( 𝑦𝐴 ) ↔ 𝑥 ∈ { ( 𝑌𝐴 ) } ) )
10 3 9 sylan9bbr ( ( 𝑌𝑉𝐴𝑊 ) → ( 𝑥 ∈ ( { 𝑌 } ↾t 𝐴 ) ↔ 𝑥 ∈ { ( 𝑌𝐴 ) } ) )
11 10 eqrdv ( ( 𝑌𝑉𝐴𝑊 ) → ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌𝐴 ) } )