Theorem resstset

Description: TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015)

	Ref	Expression
Hypotheses	resstset.1	⊢ 𝐻 = ( 𝐺 ↾s 𝐴 )
	resstset.2	⊢ 𝐽 = ( TopSet ‘ 𝐺 )
Assertion	resstset	⊢ ( 𝐴 ∈ 𝑉 → 𝐽 = ( TopSet ‘ 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		resstset.1	⊢ 𝐻 = ( 𝐺 ↾s 𝐴 )
2		resstset.2	⊢ 𝐽 = ( TopSet ‘ 𝐺 )
3		tsetid	⊢ TopSet = Slot ( TopSet ‘ ndx )
4		tsetndxnbasendx	⊢ ( TopSet ‘ ndx ) ≠ ( Base ‘ ndx )
5	1 2 3 4	resseqnbas	⊢ ( 𝐴 ∈ 𝑉 → 𝐽 = ( TopSet ‘ 𝐻 ) )