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Metamath Proof Explorer

Theorem wunress

Description: Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017) (Proof shortened by AV, 28-Oct-2024)

Ref Expression
Hypotheses wunress.1 
|- ( ph -> U e. WUni )
wunress.2 
|- ( ph -> _om e. U )
wunress.3 
|- ( ph -> W e. U )
Assertion wunress 
|- ( ph -> ( W |`s A ) e. U )

Proof

Step Hyp Ref Expression
1 wunress.1 
 |-  ( ph -> U e. WUni )
2 wunress.2 
 |-  ( ph -> _om e. U )
3 wunress.3 
 |-  ( ph -> W e. U )
4 eqid 
 |-  ( W |`s A ) = ( W |`s A )
5 eqid 
 |-  ( Base ` W ) = ( Base ` W )
6 4 5 ressval 
 |-  ( ( W e. U /\ A e. _V ) -> ( W |`s A ) = if ( ( Base ` W ) C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
7 3 6 sylan 
 |-  ( ( ph /\ A e. _V ) -> ( W |`s A ) = if ( ( Base ` W ) C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
8 1 2 basndxelwund 
 |-  ( ph -> ( Base ` ndx ) e. U )
9 incom 
 |-  ( A i^i ( Base ` W ) ) = ( ( Base ` W ) i^i A )
10 baseid 
 |-  Base = Slot ( Base ` ndx )
11 10 1 3 wunstr 
 |-  ( ph -> ( Base ` W ) e. U )
12 1 11 wunin 
 |-  ( ph -> ( ( Base ` W ) i^i A ) e. U )
13 9 12 eqeltrid 
 |-  ( ph -> ( A i^i ( Base ` W ) ) e. U )
14 1 8 13 wunop 
 |-  ( ph -> <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. e. U )
15 1 3 14 wunsets 
 |-  ( ph -> ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) e. U )
16 3 15 ifcld 
 |-  ( ph -> if ( ( Base ` W ) C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) e. U )
17 16 adantr 
 |-  ( ( ph /\ A e. _V ) -> if ( ( Base ` W ) C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) e. U )
18 7 17 eqeltrd 
 |-  ( ( ph /\ A e. _V ) -> ( W |`s A ) e. U )
19 18 ex 
 |-  ( ph -> ( A e. _V -> ( W |`s A ) e. U ) )
20 1 wun0 
 |-  ( ph -> (/) e. U )
21 reldmress 
 |-  Rel dom |`s
22 21 ovprc2 
 |-  ( -. A e. _V -> ( W |`s A ) = (/) )
23 22 eleq1d 
 |-  ( -. A e. _V -> ( ( W |`s A ) e. U <-> (/) e. U ) )
24 20 23 syl5ibrcom 
 |-  ( ph -> ( -. A e. _V -> ( W |`s A ) e. U ) )
25 19 24 pm2.61d 
 |-  ( ph -> ( W |`s A ) e. U )