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Metamath Proof Explorer

Theorem wundm

Description: A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses wun0.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
wunop.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion wundm ( 𝜑 → dom 𝐴 ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 wun0.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
2 wunop.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3 1 2 wununi ⊢ ( 𝜑 → ∪ 𝐴 ∈ 𝑈 )
4 1 3 wununi ⊢ ( 𝜑 → ∪ ∪ 𝐴 ∈ 𝑈 )
5 ssun1 ⊢ dom 𝐴 ⊆ ( dom 𝐴 ∪ ran 𝐴 )
6 dmrnssfld ⊢ ( dom 𝐴 ∪ ran 𝐴 ) ⊆ ∪ ∪ 𝐴
7 5 6 sstri ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴
8 7 a1i ⊢ ( 𝜑 → dom 𝐴 ⊆ ∪ ∪ 𝐴 )
9 1 4 8 wunss ⊢ ( 𝜑 → dom 𝐴 ∈ 𝑈 )