Theorem restutopopn 20476

Description: The restriction of the topology induced by an uniform structure to an open set. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
restutopopn	|- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( (unifTop ` U ) ↾t A ) = (unifTop ` ( U ↾t ( A X. A ) ) ) )

Proof of Theorem restutopopn

Dummy variables a b t u w x v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elutop 20471	. . . 4 |- ( U e. (UnifOn ` X ) -> ( A e. (unifTop ` U ) <-> ( A C_ X /\ A. x e. A E. t e. U ( t " { x } ) C_ A ) ) )
2	1	simprbda 623	. . 3 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> A C_ X )
3		restutop 20475	. . 3 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( (unifTop ` U ) ↾t A ) C_ (unifTop ` ( U ↾t ( A X. A ) ) ) )
4	2, 3	syldan 470	. 2 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( (unifTop ` U ) ↾t A ) C_ (unifTop ` ( U ↾t ( A X. A ) ) ) )
5		trust 20467	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
6	2, 5	syldan 470	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
7		elutop 20471	. . . . . . . . . 10 |- ( ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) -> ( b e. (unifTop ` ( U ↾t ( A X. A ) ) ) <-> ( b C_ A /\ A. x e. b E. u e. ( U ↾t ( A X. A ) ) ( u " { x } ) C_ b ) ) )
8	6, 7	syl 16	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( b e. (unifTop ` ( U ↾t ( A X. A ) ) ) <-> ( b C_ A /\ A. x e. b E. u e. ( U ↾t ( A X. A ) ) ( u " { x } ) C_ b ) ) )
9	8	simprbda 623	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b C_ A )
10	2	adantr 465	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> A C_ X )
11	9, 10	sstrd 3514	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b C_ X )
12		simp-9l 775	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> U e. (UnifOn ` X ) )
13		simplr 754	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> t e. U )
14		simp-4r 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> w e. U )
15		ustincl 20445	. . . . . . . . . . . . 13 |- ( ( U e. (UnifOn ` X ) /\ t e. U /\ w e. U ) -> ( t i^i w ) e. U )
16	12, 13, 14, 15	syl3anc 1228	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> ( t i^i w ) e. U )
17		inimass 5420	. . . . . . . . . . . . 13 |- ( ( t i^i w ) " { x } ) C_ ( ( t " { x } ) i^i ( w " { x } ) )
18		ssrin 3723	. . . . . . . . . . . . . . . 16 |- ( ( t " { x } ) C_ A -> ( ( t " { x } ) i^i ( w " { x } ) ) C_ ( A i^i ( w " { x } ) ) )
19	18	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> ( ( t " { x } ) i^i ( w " { x } ) ) C_ ( A i^i ( w " { x } ) ) )
20		simpllr 758	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> u = ( w i^i ( A X. A ) ) )
21	20	imaeq1d 5334	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> ( u " { x } ) = ( ( w i^i ( A X. A ) ) " { x } ) )
22	9	ad5antr 733	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) -> b C_ A )
23		simp-5r 768	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) -> x e. b )
24	22, 23	sseldd 3505	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) -> x e. A )
25	24	ad2antrr 725	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> x e. A )
26		vex 3116	. . . . . . . . . . . . . . . . . . . 20 |- x e. _V
27		inimasn 5421	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. _V -> ( ( w i^i ( A X. A ) ) " { x } ) = ( ( w " { x } ) i^i ( ( A X. A ) " { x } ) ) )
28	26, 27	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- ( ( w i^i ( A X. A ) ) " { x } ) = ( ( w " { x } ) i^i ( ( A X. A ) " { x } ) )
29		disjsn 4088	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( A i^i { x } ) = (/) <-> -. x e. A )
30	29	bicomi 202	. . . . . . . . . . . . . . . . . . . . . 22 |- ( -. x e. A <-> ( A i^i { x } ) = (/) )
31	30	necon1abii 2729	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A i^i { x } ) =/= (/) <-> x e. A )
32		xpima2 5449	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A i^i { x } ) =/= (/) -> ( ( A X. A ) " { x } ) = A )
33	31, 32	sylbir 213	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. A -> ( ( A X. A ) " { x } ) = A )
34	33	ineq2d 3700	. . . . . . . . . . . . . . . . . . 19 |- ( x e. A -> ( ( w " { x } ) i^i ( ( A X. A ) " { x } ) ) = ( ( w " { x } ) i^i A ) )
35	28, 34	syl5eq 2520	. . . . . . . . . . . . . . . . . 18 |- ( x e. A -> ( ( w i^i ( A X. A ) ) " { x } ) = ( ( w " { x } ) i^i A ) )
36		incom 3691	. . . . . . . . . . . . . . . . . 18 |- ( ( w " { x } ) i^i A ) = ( A i^i ( w " { x } ) )
37	35, 36	syl6eq 2524	. . . . . . . . . . . . . . . . 17 |- ( x e. A -> ( ( w i^i ( A X. A ) ) " { x } ) = ( A i^i ( w " { x } ) ) )
38	25, 37	syl 16	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> ( ( w i^i ( A X. A ) ) " { x } ) = ( A i^i ( w " { x } ) ) )
39	21, 38	eqtrd 2508	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> ( u " { x } ) = ( A i^i ( w " { x } ) ) )
40	19, 39	sseqtr4d 3541	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> ( ( t " { x } ) i^i ( w " { x } ) ) C_ ( u " { x } ) )
41		simp-5r 768	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> ( u " { x } ) C_ b )
42	40, 41	sstrd 3514	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> ( ( t " { x } ) i^i ( w " { x } ) ) C_ b )
43	17, 42	syl5ss 3515	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> ( ( t i^i w ) " { x } ) C_ b )
44		imaeq1 5330	. . . . . . . . . . . . . 14 |- ( v = ( t i^i w ) -> ( v " { x } ) = ( ( t i^i w ) " { x } ) )
45	44	sseq1d 3531	. . . . . . . . . . . . 13 |- ( v = ( t i^i w ) -> ( ( v " { x } ) C_ b <-> ( ( t i^i w ) " { x } ) C_ b ) )
46	45	rspcev 3214	. . . . . . . . . . . 12 |- ( ( ( t i^i w ) e. U /\ ( ( t i^i w ) " { x } ) C_ b ) -> E. v e. U ( v " { x } ) C_ b )
47	16, 43, 46	syl2anc 661	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) /\ t e. U ) /\ ( t " { x } ) C_ A ) -> E. v e. U ( v " { x } ) C_ b )
48		simp-4l 765	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) -> ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) )
49	48	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) -> ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) )
50	1	simplbda 624	. . . . . . . . . . . . 13 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> A. x e. A E. t e. U ( t " { x } ) C_ A )
51	50	r19.21bi 2833	. . . . . . . . . . . 12 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ x e. A ) -> E. t e. U ( t " { x } ) C_ A )
52	49, 24, 51	syl2anc 661	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) -> E. t e. U ( t " { x } ) C_ A )
53	47, 52	r19.29a 3003	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) /\ w e. U ) /\ u = ( w i^i ( A X. A ) ) ) -> E. v e. U ( v " { x } ) C_ b )
54		simplr 754	. . . . . . . . . . 11 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) -> u e. ( U ↾t ( A X. A ) ) )
55		xpexg 6709	. . . . . . . . . . . . . 14 |- ( ( A e. (unifTop ` U ) /\ A e. (unifTop ` U ) ) -> ( A X. A ) e. _V )
56	55	anidms 645	. . . . . . . . . . . . 13 |- ( A e. (unifTop ` U ) -> ( A X. A ) e. _V )
57		elrest 14679	. . . . . . . . . . . . 13 |- ( ( U e. (UnifOn ` X ) /\ ( A X. A ) e. _V ) -> ( u e. ( U ↾t ( A X. A ) ) <-> E. w e. U u = ( w i^i ( A X. A ) ) ) )
58	56, 57	sylan2 474	. . . . . . . . . . . 12 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( u e. ( U ↾t ( A X. A ) ) <-> E. w e. U u = ( w i^i ( A X. A ) ) ) )
59	58	biimpa 484	. . . . . . . . . . 11 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ u e. ( U ↾t ( A X. A ) ) ) -> E. w e. U u = ( w i^i ( A X. A ) ) )
60	48, 54, 59	syl2anc 661	. . . . . . . . . 10 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) -> E. w e. U u = ( w i^i ( A X. A ) ) )
61	53, 60	r19.29a 3003	. . . . . . . . 9 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) /\ u e. ( U ↾t ( A X. A ) ) ) /\ ( u " { x } ) C_ b ) -> E. v e. U ( v " { x } ) C_ b )
62	8	simplbda 624	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> A. x e. b E. u e. ( U ↾t ( A X. A ) ) ( u " { x } ) C_ b )
63	62	r19.21bi 2833	. . . . . . . . 9 |- ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) -> E. u e. ( U ↾t ( A X. A ) ) ( u " { x } ) C_ b )
64	61, 63	r19.29a 3003	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) /\ x e. b ) -> E. v e. U ( v " { x } ) C_ b )
65	64	ralrimiva 2878	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> A. x e. b E. v e. U ( v " { x } ) C_ b )
66		elutop 20471	. . . . . . . 8 |- ( U e. (UnifOn ` X ) -> ( b e. (unifTop ` U ) <-> ( b C_ X /\ A. x e. b E. v e. U ( v " { x } ) C_ b ) ) )
67	66	ad2antrr 725	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> ( b e. (unifTop ` U ) <-> ( b C_ X /\ A. x e. b E. v e. U ( v " { x } ) C_ b ) ) )
68	11, 65, 67	mpbir2and 920	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b e. (unifTop ` U ) )
69		df-ss 3490	. . . . . . . 8 |- ( b C_ A <-> ( b i^i A ) = b )
70	9, 69	sylib 196	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> ( b i^i A ) = b )
71	70	eqcomd 2475	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b = ( b i^i A ) )
72		ineq1 3693	. . . . . . . 8 |- ( a = b -> ( a i^i A ) = ( b i^i A ) )
73	72	eqeq2d 2481	. . . . . . 7 |- ( a = b -> ( b = ( a i^i A ) <-> b = ( b i^i A ) ) )
74	73	rspcev 3214	. . . . . 6 |- ( ( b e. (unifTop ` U ) /\ b = ( b i^i A ) ) -> E. a e. (unifTop ` U ) b = ( a i^i A ) )
75	68, 71, 74	syl2anc 661	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> E. a e. (unifTop ` U ) b = ( a i^i A ) )
76		fvex 5874	. . . . . . 7 |- (unifTop ` U ) e. _V
77		elrest 14679	. . . . . . 7 |- ( ( (unifTop ` U ) e. _V /\ A e. (unifTop ` U ) ) -> ( b e. ( (unifTop ` U ) ↾t A ) <-> E. a e. (unifTop ` U ) b = ( a i^i A ) ) )
78	76, 77	mpan 670	. . . . . 6 |- ( A e. (unifTop ` U ) -> ( b e. ( (unifTop ` U ) ↾t A ) <-> E. a e. (unifTop ` U ) b = ( a i^i A ) ) )
79	78	ad2antlr 726	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> ( b e. ( (unifTop ` U ) ↾t A ) <-> E. a e. (unifTop ` U ) b = ( a i^i A ) ) )
80	75, 79	mpbird 232	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b e. ( (unifTop ` U ) ↾t A ) )
81	80	ex 434	. . 3 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( b e. (unifTop ` ( U ↾t ( A X. A ) ) ) -> b e. ( (unifTop ` U ) ↾t A ) ) )
82	81	ssrdv 3510	. 2 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> (unifTop ` ( U ↾t ( A X. A ) ) ) C_ ( (unifTop ` U ) ↾t A ) )
83	4, 82	eqssd 3521	1 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( (unifTop ` U ) ↾t A ) = (unifTop ` ( U ↾t ( A X. A ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113   i^i cin 3475   C_ wss 3476   (/)c0 3785   {csn 4027   X. cxp 4997   "cima 5002   ` cfv 5586  (class class class)co 6282   ↾_t crest 14672  UnifOncust 20437  unifTopcutop 20468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-rest 14674  df-ust 20438  df-utop 20469

This theorem is referenced by:  ressusp  20503