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Metamath Proof Explorer

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 )