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Metamath Proof Explorer

Theorem ustbas2

Description: Second direction for ustbas . (Contributed by Thierry Arnoux, 16-Nov-2017)

Ref Expression
Assertion ustbas2 
|- ( U e. ( UnifOn ` X ) -> X = dom U. U )

Proof

Step Hyp Ref Expression
1 dmxpid 
 |-  dom ( X X. X ) = X
2 ustbasel 
 |-  ( U e. ( UnifOn ` X ) -> ( X X. X ) e. U )
3 elssuni 
 |-  ( ( X X. X ) e. U -> ( X X. X ) C_ U. U )
4 2 3 syl 
 |-  ( U e. ( UnifOn ` X ) -> ( X X. X ) C_ U. U )
5 elfvex 
 |-  ( U e. ( UnifOn ` X ) -> X e. _V )
6 isust 
 |-  ( X e. _V -> ( U e. ( UnifOn ` X ) <-> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) ) )
7 5 6 syl 
 |-  ( U e. ( UnifOn ` X ) -> ( U e. ( UnifOn ` X ) <-> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) ) )
8 7 ibi 
 |-  ( U e. ( UnifOn ` X ) -> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) )
9 8 simp1d 
 |-  ( U e. ( UnifOn ` X ) -> U C_ ~P ( X X. X ) )
10 9 unissd 
 |-  ( U e. ( UnifOn ` X ) -> U. U C_ U. ~P ( X X. X ) )
11 unipw 
 |-  U. ~P ( X X. X ) = ( X X. X )
12 10 11 sseqtrdi 
 |-  ( U e. ( UnifOn ` X ) -> U. U C_ ( X X. X ) )
13 4 12 eqssd 
 |-  ( U e. ( UnifOn ` X ) -> ( X X. X ) = U. U )
14 13 dmeqd 
 |-  ( U e. ( UnifOn ` X ) -> dom ( X X. X ) = dom U. U )
15 1 14 eqtr3id 
 |-  ( U e. ( UnifOn ` X ) -> X = dom U. U )