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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 )