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Metamath Proof Explorer

Theorem wunfv

Description: A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017)

Ref Expression
Hypotheses wun0.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
wunop.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion wunfv ( 𝜑 → ( 𝐴 ‘ 𝐵 ) ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 wun0.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
2 wunop.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3 1 2 wunrn ⊢ ( 𝜑 → ran 𝐴 ∈ 𝑈 )
4 1 3 wununi ⊢ ( 𝜑 → ∪ ran 𝐴 ∈ 𝑈 )
5 fvssunirn ⊢ ( 𝐴 ‘ 𝐵 ) ⊆ ∪ ran 𝐴
6 5 a1i ⊢ ( 𝜑 → ( 𝐴 ‘ 𝐵 ) ⊆ ∪ ran 𝐴 )
7 1 4 6 wunss ⊢ ( 𝜑 → ( 𝐴 ‘ 𝐵 ) ∈ 𝑈 )