Theorem restutopopn 22042

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 22037	. . . 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 653	. . 3 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> A C_ X )
3		restutop 22041	. . 3 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( (unifTop ` U ) ↾t A ) C_ (unifTop ` ( U ↾t ( A X. A ) ) ) )
4	2, 3	syldan 487	. 2 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( (unifTop ` U ) ↾t A ) C_ (unifTop ` ( U ↾t ( A X. A ) ) ) )
5		trust 22033	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
6	2, 5	syldan 487	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
7		elutop 22037	. . . . . . . . . 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 653	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b C_ A )
10	2	adantr 481	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> A C_ X )
11	9, 10	sstrd 3613	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b C_ X )
12		simp-9l 816	. . . . . . . . . . . . 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 792	. . . . . . . . . . . . 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 807	. . . . . . . . . . . . 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 22011	. . . . . . . . . . . . 13 |- ( ( U e. (UnifOn ` X ) /\ t e. U /\ w e. U ) -> ( t i^i w ) e. U )
16	12, 13, 14, 15	syl3anc 1326	. . . . . . . . . . . 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 5549	. . . . . . . . . . . . 13 |- ( ( t i^i w ) " { x } ) C_ ( ( t " { x } ) i^i ( w " { x } ) )
18		ssrin 3838	. . . . . . . . . . . . . . . 16 |- ( ( t " { x } ) C_ A -> ( ( t " { x } ) i^i ( w " { x } ) ) C_ ( A i^i ( w " { x } ) ) )
19	18	adantl 482	. . . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . . . . 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 5465	. . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . 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 809	. . . . . . . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . 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 3203	. . . . . . . . . . . . . . . . . . . 20 |- x e. _V
27		inimasn 5550	. . . . . . . . . . . . . . . . . . . 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 5579	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. A -> ( ( A X. A ) " { x } ) = A )
30	29	ineq2d 3814	. . . . . . . . . . . . . . . . . . 19 |- ( x e. A -> ( ( w " { x } ) i^i ( ( A X. A ) " { x } ) ) = ( ( w " { x } ) i^i A ) )
31	28, 30	syl5eq 2668	. . . . . . . . . . . . . . . . . 18 |- ( x e. A -> ( ( w i^i ( A X. A ) ) " { x } ) = ( ( w " { x } ) i^i A ) )
32		incom 3805	. . . . . . . . . . . . . . . . . 18 |- ( ( w " { x } ) i^i A ) = ( A i^i ( w " { x } ) )
33	31, 32	syl6eq 2672	. . . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . . 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 3642	. . . . . . . . . . . . . 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 809	. . . . . . . . . . . . . 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 3613	. . . . . . . . . . . . 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 3614	. . . . . . . . . . . 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 5461	. . . . . . . . . . . . . 14 |- ( v = ( t i^i w ) -> ( v " { x } ) = ( ( t i^i w ) " { x } ) )
41	40	sseq1d 3632	. . . . . . . . . . . . 13 |- ( v = ( t i^i w ) -> ( ( v " { x } ) C_ b <-> ( ( t i^i w ) " { x } ) C_ b ) )
42	41	rspcev 3309	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 806	. . . . . . . . . . . . 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 762	. . . . . . . . . . . 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 654	. . . . . . . . . . . . 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 2932	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 3078	. . . . . . . . . 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 792	. . . . . . . . . . 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 6963	. . . . . . . . . . . . 13 |- ( A e. (unifTop ` U ) -> ( A X. A ) e. _V )
52		elrest 16088	. . . . . . . . . . . . 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 491	. . . . . . . . . . . 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 501	. . . . . . . . . . 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 693	. . . . . . . . . 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 3078	. . . . . . . . 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 654	. . . . . . . . . 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 2932	. . . . . . . . 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 3078	. . . . . . . 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 2966	. . . . . . 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 22037	. . . . . . . 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 762	. . . . . . 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 957	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b e. (unifTop ` U ) )
64		df-ss 3588	. . . . . . . 8 |- ( b C_ A <-> ( b i^i A ) = b )
65	9, 64	sylib 208	. . . . . . 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 2628	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) /\ b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) -> b = ( b i^i A ) )
67		ineq1 3807	. . . . . . . 8 |- ( a = b -> ( a i^i A ) = ( b i^i A ) )
68	67	eqeq2d 2632	. . . . . . 7 |- ( a = b -> ( b = ( a i^i A ) <-> b = ( b i^i A ) ) )
69	68	rspcev 3309	. . . . . 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 693	. . . . 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 6201	. . . . . . 7 |- (unifTop ` U ) e. _V
72		elrest 16088	. . . . . . 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 706	. . . . . 6 |- ( A e. (unifTop ` U ) -> ( b e. ( (unifTop ` U ) ↾t A ) <-> E. a e. (unifTop ` U ) b = ( a i^i A ) ) )
74	73	ad2antlr 763	. . . . 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 247	. . . 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 450	. . 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 3609	. 2 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> (unifTop ` ( U ↾t ( A X. A ) ) ) C_ ( (unifTop ` U ) ↾t A ) )
78	4, 77	eqssd 3620	1 |- ( ( U e. (UnifOn ` X ) /\ A e. (unifTop ` U ) ) -> ( (unifTop ` U ) ↾t A ) = (unifTop ` ( U ↾t ( A X. A ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   X. cxp 5112   "cima 5117   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081  UnifOncust 22003  unifTopcutop 22034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-rest 16083  df-ust 22004  df-utop 22035

This theorem is referenced by:  ressusp  22069