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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 A ∈ U
6 1 5 wunelss ⊢ φ → B A ⊆ U
7 3 2 elmapd ⊢ φ → F ∈ B A ↔ F : A ⟶ B
8 4 7 mpbird ⊢ φ → F ∈ B A
9 6 8 sseldd ⊢ φ → F ∈ U