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Metamath Proof Explorer

Theorem wunmap

Description: A weak universe is closed under mappings. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses wun0.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
wunop.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
wunop.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )
Assertion wunmap ( 𝜑 → ( 𝐴 ↑m 𝐵 ) ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 wun0.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
2 wunop.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3 wunop.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )
4 1 2 3 wunpm ⊢ ( 𝜑 → ( 𝐴 ↑pm 𝐵 ) ∈ 𝑈 )
5 mapsspm ⊢ ( 𝐴 ↑m 𝐵 ) ⊆ ( 𝐴 ↑pm 𝐵 )
6 5 a1i ⊢ ( 𝜑 → ( 𝐴 ↑m 𝐵 ) ⊆ ( 𝐴 ↑pm 𝐵 ) )
7 1 4 6 wunss ⊢ ( 𝜑 → ( 𝐴 ↑m 𝐵 ) ∈ 𝑈 )