Theorem wuntpos

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

	Ref	Expression
Hypotheses	wun0.1	|- ( ph -> U e. WUni )
	wunop.2	|- ( ph -> A e. U )
Assertion	wuntpos	|- ( ph -> tpos A e. U )

Proof

Step	Hyp	Ref	Expression
1		wun0.1	|- ( ph -> U e. WUni )
2		wunop.2	|- ( ph -> A e. U )
3	1 2	wundm	|- ( ph -> dom A e. U )
4	1 3	wuncnv	|- ( ph -> `' dom A e. U )
5	1	wun0	|- ( ph -> (/) e. U )
6	1 5	wunsn	|- ( ph -> { (/) } e. U )
7	1 4 6	wunun	|- ( ph -> ( `' dom A u. { (/) } ) e. U )
8	1 2	wunrn	|- ( ph -> ran A e. U )
9	1 7 8	wunxp	|- ( ph -> ( ( `' dom A u. { (/) } ) X. ran A ) e. U )
10		tposssxp	|- tpos A C_ ( ( `' dom A u. { (/) } ) X. ran A )
11	10	a1i	|- ( ph -> tpos A C_ ( ( `' dom A u. { (/) } ) X. ran A ) )
12	1 9 11	wunss	|- ( ph -> tpos A e. U )