Theorem wuntpos

Description: A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017)

	Ref	Expression
Hypotheses	wun0.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wunop.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion	wuntpos	⊢ ( 𝜑 → tpos 𝐴 ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		wun0.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
2		wunop.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3	1 2	wundm	⊢ ( 𝜑 → dom 𝐴 ∈ 𝑈 )
4	1 3	wuncnv	⊢ ( 𝜑 → ◡ dom 𝐴 ∈ 𝑈 )
5	1	wun0	⊢ ( 𝜑 → ∅ ∈ 𝑈 )
6	1 5	wunsn	⊢ ( 𝜑 → { ∅ } ∈ 𝑈 )
7	1 4 6	wunun	⊢ ( 𝜑 → ( ◡ dom 𝐴 ∪ { ∅ } ) ∈ 𝑈 )
8	1 2	wunrn	⊢ ( 𝜑 → ran 𝐴 ∈ 𝑈 )
9	1 7 8	wunxp	⊢ ( 𝜑 → ( ( ◡ dom 𝐴 ∪ { ∅ } ) × ran 𝐴 ) ∈ 𝑈 )
10		tposssxp	⊢ tpos 𝐴 ⊆ ( ( ◡ dom 𝐴 ∪ { ∅ } ) × ran 𝐴 )
11	10	a1i	⊢ ( 𝜑 → tpos 𝐴 ⊆ ( ( ◡ dom 𝐴 ∪ { ∅ } ) × ran 𝐴 ) )
12	1 9 11	wunss	⊢ ( 𝜑 → tpos 𝐴 ∈ 𝑈 )