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Metamath Proof Explorer

Theorem wunf

Description: A weak universe is closed under functions with known domain and codomain. (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 )
wunf.3 ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
Assertion wunf ( 𝜑 → 𝐹 ∈ 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 wunf.3 ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
5 1 3 2 wunmap ⊢ ( 𝜑 → ( 𝐵 ↑m 𝐴 ) ∈ 𝑈 )
6 1 5 wunelss ⊢ ( 𝜑 → ( 𝐵 ↑m 𝐴 ) ⊆ 𝑈 )
7 3 2 elmapd ⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐵 ↑m 𝐴 ) ↔ 𝐹 : 𝐴 ⟶ 𝐵 ) )
8 4 7 mpbird ⊢ ( 𝜑 → 𝐹 ∈ ( 𝐵 ↑m 𝐴 ) )
9 6 8 sseldd ⊢ ( 𝜑 → 𝐹 ∈ 𝑈 )