This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )