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Metamath Proof Explorer

Theorem wunpm

Description: A weak universe is closed under partial 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 wunpm ⊢ φ → A ↑ 𝑝𝑚 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 3 2 wunxp ⊢ φ → B × A ∈ U
5 1 4 wunpw ⊢ φ → 𝒫 B × A ∈ U
6 pmsspw ⊢ A ↑ 𝑝𝑚 B ⊆ 𝒫 B × A
7 6 a1i ⊢ φ → A ↑ 𝑝𝑚 B ⊆ 𝒫 B × A
8 1 5 7 wunss ⊢ φ → A ↑ 𝑝𝑚 B ∈ U