Theorem bj-restsnid

Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn and bj-restsnss . (Contributed by BJ, 27-Apr-2021)

	Ref	Expression
Assertion	bj-restsnid	⊢ ( { 𝐴 } ↾t 𝐴 ) = { 𝐴 }

Proof

Step	Hyp	Ref	Expression
1		ssid	⊢ 𝐴 ⊆ 𝐴
2		bj-restsnss	⊢ ( ( 𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴 ) → ( { 𝐴 } ↾t 𝐴 ) = { 𝐴 } )
3	1 2	mpan2	⊢ ( 𝐴 ∈ V → ( { 𝐴 } ↾t 𝐴 ) = { 𝐴 } )
4		df-rest	⊢ ↾t = ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ran ( 𝑧 ∈ 𝑥 ↦ ( 𝑧 ∩ 𝑦 ) ) )
5	4	reldmmpo	⊢ Rel dom ↾t
6	5	ovprc2	⊢ ( ¬ 𝐴 ∈ V → ( { 𝐴 } ↾t 𝐴 ) = ∅ )
7		snprc	⊢ ( ¬ 𝐴 ∈ V ↔ { 𝐴 } = ∅ )
8	7	biimpi	⊢ ( ¬ 𝐴 ∈ V → { 𝐴 } = ∅ )
9	6 8	eqtr4d	⊢ ( ¬ 𝐴 ∈ V → ( { 𝐴 } ↾t 𝐴 ) = { 𝐴 } )
10	3 9	pm2.61i	⊢ ( { 𝐴 } ↾t 𝐴 ) = { 𝐴 }