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 ( 𝜑𝑈 ∈ WUni )
wunop.2 ( 𝜑𝐴𝑈 )
wunop.3 ( 𝜑𝐵𝑈 )
Assertion wunpm ( 𝜑 → ( 𝐴pm 𝐵 ) ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 wun0.1 ( 𝜑𝑈 ∈ WUni )
2 wunop.2 ( 𝜑𝐴𝑈 )
3 wunop.3 ( 𝜑𝐵𝑈 )
4 1 3 2 wunxp ( 𝜑 → ( 𝐵 × 𝐴 ) ∈ 𝑈 )
5 1 4 wunpw ( 𝜑 → 𝒫 ( 𝐵 × 𝐴 ) ∈ 𝑈 )
6 pmsspw ( 𝐴pm 𝐵 ) ⊆ 𝒫 ( 𝐵 × 𝐴 )
7 6 a1i ( 𝜑 → ( 𝐴pm 𝐵 ) ⊆ 𝒫 ( 𝐵 × 𝐴 ) )
8 1 5 7 wunss ( 𝜑 → ( 𝐴pm 𝐵 ) ∈ 𝑈 )