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 )