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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