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Metamath Proof Explorer

Theorem wunrn

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

Ref Expression
Hypotheses wun0.1 ⊢ φ → U ∈ WUni
wunop.2 ⊢ φ → A ∈ U
Assertion wunrn ⊢ φ → ran ⁡ 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 ssun2 ⊢ ran ⁡ A ⊆ dom ⁡ A ∪ ran ⁡ A
6 dmrnssfld ⊢ dom ⁡ A ∪ ran ⁡ A ⊆ ⋃ ⋃ A
7 5 6 sstri ⊢ ran ⁡ A ⊆ ⋃ ⋃ A
8 7 a1i ⊢ φ → ran ⁡ A ⊆ ⋃ ⋃ A
9 1 4 8 wunss ⊢ φ → ran ⁡ A ∈ U