Theorem toponrestid

Description: Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022)

	Ref	Expression
Hypothesis	toponrestid.t	⊢ 𝐴 ∈ ( TopOn ‘ 𝐵 )
Assertion	toponrestid	⊢ 𝐴 = ( 𝐴 ↾t 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		toponrestid.t	⊢ 𝐴 ∈ ( TopOn ‘ 𝐵 )
2	1	toponunii	⊢ 𝐵 = ∪ 𝐴
3	2	restid	⊢ ( 𝐴 ∈ ( TopOn ‘ 𝐵 ) → ( 𝐴 ↾t 𝐵 ) = 𝐴 )
4	1 3	ax-mp	⊢ ( 𝐴 ↾t 𝐵 ) = 𝐴
5	4	eqcomi	⊢ 𝐴 = ( 𝐴 ↾t 𝐵 )