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Metamath Proof Explorer

Theorem wunco

Description: A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017)

Ref Expression
Hypotheses wun0.1 ⊢ φ → U ∈ WUni
wunop.2 ⊢ φ → A ∈ U
wunco.3 ⊢ φ → B ∈ U
Assertion wunco ⊢ φ → A ∘ B ∈ U

Proof

Step Hyp Ref Expression
1 wun0.1 ⊢ φ → U ∈ WUni
2 wunop.2 ⊢ φ → A ∈ U
3 wunco.3 ⊢ φ → B ∈ U
4 1 3 wundm ⊢ φ → dom ⁡ B ∈ U
5 dmcoss ⊢ dom ⁡ A ∘ B ⊆ dom ⁡ B
6 5 a1i ⊢ φ → dom ⁡ A ∘ B ⊆ dom ⁡ B
7 1 4 6 wunss ⊢ φ → dom ⁡ A ∘ B ∈ U
8 1 2 wunrn ⊢ φ → ran ⁡ A ∈ U
9 rncoss ⊢ ran ⁡ A ∘ B ⊆ ran ⁡ A
10 9 a1i ⊢ φ → ran ⁡ A ∘ B ⊆ ran ⁡ A
11 1 8 10 wunss ⊢ φ → ran ⁡ A ∘ B ∈ U
12 1 7 11 wunxp ⊢ φ → dom ⁡ A ∘ B × ran ⁡ A ∘ B ∈ U
13 relco ⊢ Rel ⁡ A ∘ B
14 relssdmrn ⊢ Rel ⁡ A ∘ B → A ∘ B ⊆ dom ⁡ A ∘ B × ran ⁡ A ∘ B
15 13 14 mp1i ⊢ φ → A ∘ B ⊆ dom ⁡ A ∘ B × ran ⁡ A ∘ B
16 1 12 15 wunss ⊢ φ → A ∘ B ∈ U