Theorem wunstr

Description: Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017)

	Ref	Expression
Hypotheses	strfvss.e	⊢ 𝐸 = Slot 𝑁
	wunstr.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wunstr.s	⊢ ( 𝜑 → 𝑆 ∈ 𝑈 )
Assertion	wunstr	⊢ ( 𝜑 → ( 𝐸 ‘ 𝑆 ) ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		strfvss.e	⊢ 𝐸 = Slot 𝑁
2		wunstr.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
3		wunstr.s	⊢ ( 𝜑 → 𝑆 ∈ 𝑈 )
4	2 3	wunrn	⊢ ( 𝜑 → ran 𝑆 ∈ 𝑈 )
5	2 4	wununi	⊢ ( 𝜑 → ∪ ran 𝑆 ∈ 𝑈 )
6	1	strfvss	⊢ ( 𝐸 ‘ 𝑆 ) ⊆ ∪ ran 𝑆
7	6	a1i	⊢ ( 𝜑 → ( 𝐸 ‘ 𝑆 ) ⊆ ∪ ran 𝑆 )
8	2 5 7	wunss	⊢ ( 𝜑 → ( 𝐸 ‘ 𝑆 ) ∈ 𝑈 )