Metamath Proof Explorer


Theorem bj-restsnss

Description: Special case of bj-restsn . (Contributed by BJ, 27-Apr-2021)

Ref Expression
Assertion bj-restsnss ( ( 𝑌𝑉𝐴𝑌 ) → ( { 𝑌 } ↾t 𝐴 ) = { 𝐴 } )

Proof

Step Hyp Ref Expression
1 sseqin2 ( 𝐴𝑌 ↔ ( 𝑌𝐴 ) = 𝐴 )
2 sneq ( ( 𝑌𝐴 ) = 𝐴 → { ( 𝑌𝐴 ) } = { 𝐴 } )
3 1 2 sylbi ( 𝐴𝑌 → { ( 𝑌𝐴 ) } = { 𝐴 } )
4 ssexg ( ( 𝐴𝑌𝑌𝑉 ) → 𝐴 ∈ V )
5 4 ancoms ( ( 𝑌𝑉𝐴𝑌 ) → 𝐴 ∈ V )
6 bj-restsn ( ( 𝑌𝑉𝐴 ∈ V ) → ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌𝐴 ) } )
7 5 6 syldan ( ( 𝑌𝑉𝐴𝑌 ) → ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌𝐴 ) } )
8 eqeq2 ( { ( 𝑌𝐴 ) } = { 𝐴 } → ( ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌𝐴 ) } ↔ ( { 𝑌 } ↾t 𝐴 ) = { 𝐴 } ) )
9 8 biimpa ( ( { ( 𝑌𝐴 ) } = { 𝐴 } ∧ ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌𝐴 ) } ) → ( { 𝑌 } ↾t 𝐴 ) = { 𝐴 } )
10 3 7 9 syl2an2 ( ( 𝑌𝑉𝐴𝑌 ) → ( { 𝑌 } ↾t 𝐴 ) = { 𝐴 } )