Theorem wuncnv

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

	Ref	Expression
Hypotheses	wun0.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wunop.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion	wuncnv	⊢ ( 𝜑 → ◡ 𝐴 ∈ 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 2	wundm	⊢ ( 𝜑 → dom 𝐴 ∈ 𝑈 )
5	1 3 4	wunxp	⊢ ( 𝜑 → ( ran 𝐴 × dom 𝐴 ) ∈ 𝑈 )
6		cnvssrndm	⊢ ◡ 𝐴 ⊆ ( ran 𝐴 × dom 𝐴 )
7	6	a1i	⊢ ( 𝜑 → ◡ 𝐴 ⊆ ( ran 𝐴 × dom 𝐴 ) )
8	1 5 7	wunss	⊢ ( 𝜑 → ◡ 𝐴 ∈ 𝑈 )