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 )