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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 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 B ⊆ A ↑ 𝑝𝑚 B
6 5 a1i ⊢ φ → A B ⊆ A ↑ 𝑝𝑚 B
7 1 4 6 wunss ⊢ φ → A B ∈ U