Theorem restutopopn 21253

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 21248	. . . 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 629	. . 3 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> A C_ X )
3		restutop 21252	. . 3 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( (unifTop ` U ) ↾t A ) C_ (unifTop ` ( U ↾t ( A X. A ) ) ) )
4	2, 3	syldan 473	. 2 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( (unifTop ` U ) ↾t A ) C_ (unifTop ` ( U ↾t ( A X. A ) ) ) )
5		trust 21244	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
6	2, 5	syldan 473	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
7		elutop 21248	. . . . . . . . . 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 17	. . . . . . . . 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 629	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b C_ A )
10	2	adantr 467	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> A C_ X )
11	9, 10	sstrd 3442	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b C_ X )
12		simp-9l 786	. . . . . . . . . . . . 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 762	. . . . . . . . . . . . 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 777	. . . . . . . . . . . . 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 21222	. . . . . . . . . . . . 13 |- ( ( U e. (UnifOn ` X ) /\ t e. U /\ w e. U ) -> ( t i^i w ) e. U )
16	12, 13, 14, 15	syl3anc 1268	. . . . . . . . . . . 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 5252	. . . . . . . . . . . . 13 |- ( ( t i^i w ) " { x } ) C_ ( ( t " { x } ) i^i ( w " { x } ) )
18		ssrin 3657	. . . . . . . . . . . . . . . 16 |- ( ( t " { x } ) C_ A -> ( ( t " { x } ) i^i ( w " { x } ) ) C_ ( A i^i ( w " { x } ) ) )
19	18	adantl 468	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . 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 5167	. . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . 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 3433	. . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . 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 3048	. . . . . . . . . . . . . . . . . . . 20 |- x e. _V
27		inimasn 5253	. . . . . . . . . . . . . . . . . . . 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		xpimasn 5282	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. A -> ( ( A X. A ) " { x } ) = A )
30	29	ineq2d 3634	. . . . . . . . . . . . . . . . . . 19 |- ( x e. A -> ( ( w " { x } ) i^i ( ( A X. A ) " { x } ) ) = ( ( w " { x } ) i^i A ) )
31	28, 30	syl5eq 2497	. . . . . . . . . . . . . . . . . 18 |- ( x e. A -> ( ( w i^i ( A X. A ) ) " { x } ) = ( ( w " { x } ) i^i A ) )
32		incom 3625	. . . . . . . . . . . . . . . . . 18 |- ( ( w " { x } ) i^i A ) = ( A i^i ( w " { x } ) )
33	31, 32	syl6eq 2501	. . . . . . . . . . . . . . . . 17 |- ( x e. A -> ( ( w i^i ( A X. A ) ) " { x } ) = ( A i^i ( w " { x } ) ) )
34	25, 33	syl 17	. . . . . . . . . . . . . . . 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 } ) ) )
35	21, 34	eqtrd 2485	. . . . . . . . . . . . . . 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 } ) ) )
36	19, 35	sseqtr4d 3469	. . . . . . . . . . . . . 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 } ) )
37		simp-5r 779	. . . . . . . . . . . . . 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 )
38	36, 37	sstrd 3442	. . . . . . . . . . . . 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 )
39	17, 38	syl5ss 3443	. . . . . . . . . . . 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 )
40		imaeq1 5163	. . . . . . . . . . . . . 14 |- ( v = ( t i^i w ) -> ( v " { x } ) = ( ( t i^i w ) " { x } ) )
41	40	sseq1d 3459	. . . . . . . . . . . . 13 |- ( v = ( t i^i w ) -> ( ( v " { x } ) C_ b <-> ( ( t i^i w ) " { x } ) C_ b ) )
42	41	rspcev 3150	. . . . . . . . . . . 12 |- ( ( ( t i^i w ) e. U /\ ( ( t i^i w ) " { x } ) C_ b ) -> E. v e. U ( v " { x } ) C_ b )
43	16, 39, 42	syl2anc 667	. . . . . . . . . . 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 )
44		simp-4l 776	. . . . . . . . . . . . 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 ) ) )
45	44	ad2antrr 732	. . . . . . . . . . . 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 ) ) )
46	1	simplbda 630	. . . . . . . . . . . . 13 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> A. x e. A E. t e. U ( t " { x } ) C_ A )
47	46	r19.21bi 2757	. . . . . . . . . . . 12 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ x e. A ) -> E. t e. U ( t " { x } ) C_ A )
48	45, 24, 47	syl2anc 667	. . . . . . . . . . 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 )
49	43, 48	r19.29a 2932	. . . . . . . . . 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 )
50		simplr 762	. . . . . . . . . . 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 ) ) )
51		sqxpexg 6596	. . . . . . . . . . . . 13 |- ( A e. (unifTop ` U ) -> ( A X. A ) e. _V )
52		elrest 15326	. . . . . . . . . . . . 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 ) ) ) )
53	51, 52	sylan2 477	. . . . . . . . . . . 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 ) ) ) )
54	53	biimpa 487	. . . . . . . . . . 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 ) ) )
55	44, 50, 54	syl2anc 667	. . . . . . . . . 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 ) ) )
56	49, 55	r19.29a 2932	. . . . . . . . 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 )
57	8	simplbda 630	. . . . . . . . . 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 )
58	57	r19.21bi 2757	. . . . . . . . 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 )
59	56, 58	r19.29a 2932	. . . . . . . 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 )
60	59	ralrimiva 2802	. . . . . . 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 )
61		elutop 21248	. . . . . . . 8 |- ( U e. (UnifOn ` X ) -> ( b e. (unifTop ` U ) <-> ( b C_ X /\ A. x e. b E. v e. U ( v " { x } ) C_ b ) ) )
62	61	ad2antrr 732	. . . . . . 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 ) ) )
63	11, 60, 62	mpbir2and 933	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b e. (unifTop ` U ) )
64		df-ss 3418	. . . . . . . 8 |- ( b C_ A <-> ( b i^i A ) = b )
65	9, 64	sylib 200	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> ( b i^i A ) = b )
66	65	eqcomd 2457	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b = ( b i^i A ) )
67		ineq1 3627	. . . . . . . 8 |- ( a = b -> ( a i^i A ) = ( b i^i A ) )
68	67	eqeq2d 2461	. . . . . . 7 |- ( a = b -> ( b = ( a i^i A ) <-> b = ( b i^i A ) ) )
69	68	rspcev 3150	. . . . . 6 |- ( ( b e. (unifTop ` U ) /\ b = ( b i^i A ) ) -> E. a e. (unifTop ` U ) b = ( a i^i A ) )
70	63, 66, 69	syl2anc 667	. . . . 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 ) )
71		fvex 5875	. . . . . . 7 |- (unifTop ` U ) e. _V
72		elrest 15326	. . . . . . 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 ) ) )
73	71, 72	mpan 676	. . . . . 6 |- ( A e. (unifTop ` U ) -> ( b e. ( (unifTop ` U ) ↾t A ) <-> E. a e. (unifTop ` U ) b = ( a i^i A ) ) )
74	73	ad2antlr 733	. . . . 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 ) ) )
75	70, 74	mpbird 236	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b e. ( (unifTop ` U ) ↾t A ) )
76	75	ex 436	. . 3 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( b e. (unifTop ` ( U ↾t ( A X. A ) ) ) -> b e. ( (unifTop ` U ) ↾t A ) ) )
77	76	ssrdv 3438	. 2 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> (unifTop ` ( U ↾t ( A X. A ) ) ) C_ ( (unifTop ` U ) ↾t A ) )
78	4, 77	eqssd 3449	1 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( (unifTop ` U ) ↾t A ) = (unifTop ` ( U ↾t ( A X. A ) ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   E.wrex 2738   _Vcvv 3045   i^i cin 3403   C_ wss 3404   {csn 3968   X. cxp 4832   "cima 4837   ` cfv 5582  (class class class)co 6290   ↾_t crest 15319  UnifOncust 21214  unifTopcutop 21245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-rest 15321  df-ust 21215  df-utop 21246

This theorem is referenced by:  ressusp  21280