Theorem wunress

Description: Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017)

	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		df-base	|- Base = Slot 1
9	1 2	wunndx	|- ( ph -> ndx e. U )
10	8 1 9	wunstr	|- ( ph -> ( Base ` ndx ) e. U )
11		incom	|- ( A i^i ( Base ` W ) ) = ( ( Base ` W ) i^i A )
12	8 1 3	wunstr	|- ( ph -> ( Base ` W ) e. U )
13	1 12	wunin	|- ( ph -> ( ( Base ` W ) i^i A ) e. U )
14	11 13	eqeltrid	|- ( ph -> ( A i^i ( Base ` W ) ) e. U )
15	1 10 14	wunop	|- ( ph -> <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. e. U )
16	1 3 15	wunsets	|- ( ph -> ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) e. U )
17	3 16	ifcld	|- ( ph -> if ( ( Base ` W ) C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) e. U )
18	17	adantr	|- ( ( ph /\ A e. _V ) -> if ( ( Base ` W ) C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) e. U )
19	7 18	eqeltrd	|- ( ( ph /\ A e. _V ) -> ( W |`s A ) e. U )
20	19	ex	|- ( ph -> ( A e. _V -> ( W |`s A ) e. U ) )
21	1	wun0	|- ( ph -> (/) e. U )
22		reldmress	|- Rel dom |`s
23	22	ovprc2	|- ( -. A e. _V -> ( W |`s A ) = (/) )
24	23	eleq1d	|- ( -. A e. _V -> ( ( W |`s A ) e. U <-> (/) e. U ) )
25	21 24	syl5ibrcom	|- ( ph -> ( -. A e. _V -> ( W |`s A ) e. U ) )
26	20 25	pm2.61d	|- ( ph -> ( W |`s A ) e. U )