Theorem restutop 22041

Description: Restriction of a topology induced by an uniform structure. (Contributed by Thierry Arnoux, 12-Dec-2017.)

Assertion

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

Proof of Theorem restutop

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

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) -> ( U e. (UnifOn ` X ) /\ A C_ X ) )
2		fvexd 6203	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> (unifTop ` U ) e. _V )
3		elfvex 6221	. . . . . . . . 9 |- ( U e. (UnifOn ` X ) -> X e. _V )
4	3	adantr 481	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> X e. _V )
5		simpr 477	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> A C_ X )
6	4, 5	ssexd 4805	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> A e. _V )
7		elrest 16088	. . . . . . 7 |- ( ( (unifTop ` U ) e. _V /\ A e. _V ) -> ( b e. ( (unifTop ` U ) ↾t A ) <-> E. a e. (unifTop ` U ) b = ( a i^i A ) ) )
8	2, 6, 7	syl2anc 693	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( b e. ( (unifTop ` U ) ↾t A ) <-> E. a e. (unifTop ` U ) b = ( a i^i A ) ) )
9	8	biimpa 501	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) -> E. a e. (unifTop ` U ) b = ( a i^i A ) )
10		inss2 3834	. . . . . . 7 |- ( a i^i A ) C_ A
11		sseq1 3626	. . . . . . 7 |- ( b = ( a i^i A ) -> ( b C_ A <-> ( a i^i A ) C_ A ) )
12	10, 11	mpbiri 248	. . . . . 6 |- ( b = ( a i^i A ) -> b C_ A )
13	12	rexlimivw 3029	. . . . 5 |- ( E. a e. (unifTop ` U ) b = ( a i^i A ) -> b C_ A )
14	9, 13	syl 17	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) -> b C_ A )
15		simp-5l 808	. . . . . . . . . 10 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) -> U e. (UnifOn ` X ) )
16	15	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) /\ u e. U ) /\ ( u " { x } ) C_ a ) -> U e. (UnifOn ` X ) )
17	6	ad6antr 772	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) /\ u e. U ) /\ ( u " { x } ) C_ a ) -> A e. _V )
18		xpexg 6960	. . . . . . . . . 10 |- ( ( A e. _V /\ A e. _V ) -> ( A X. A ) e. _V )
19	17, 17, 18	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) /\ u e. U ) /\ ( u " { x } ) C_ a ) -> ( A X. A ) e. _V )
20		simplr 792	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) /\ u e. U ) /\ ( u " { x } ) C_ a ) -> u e. U )
21		elrestr 16089	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ ( A X. A ) e. _V /\ u e. U ) -> ( u i^i ( A X. A ) ) e. ( U ↾t ( A X. A ) ) )
22	16, 19, 20, 21	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) /\ u e. U ) /\ ( u " { x } ) C_ a ) -> ( u i^i ( A X. A ) ) e. ( U ↾t ( A X. A ) ) )
23		inss1 3833	. . . . . . . . . . . . 13 |- ( u i^i ( A X. A ) ) C_ u
24		imass1 5500	. . . . . . . . . . . . 13 |- ( ( u i^i ( A X. A ) ) C_ u -> ( ( u i^i ( A X. A ) ) " { x } ) C_ ( u " { x } ) )
25	23, 24	ax-mp 5	. . . . . . . . . . . 12 |- ( ( u i^i ( A X. A ) ) " { x } ) C_ ( u " { x } )
26		sstr 3611	. . . . . . . . . . . 12 |- ( ( ( ( u i^i ( A X. A ) ) " { x } ) C_ ( u " { x } ) /\ ( u " { x } ) C_ a ) -> ( ( u i^i ( A X. A ) ) " { x } ) C_ a )
27	25, 26	mpan 706	. . . . . . . . . . 11 |- ( ( u " { x } ) C_ a -> ( ( u i^i ( A X. A ) ) " { x } ) C_ a )
28		imassrn 5477	. . . . . . . . . . . . . . 15 |- ( ( u i^i ( A X. A ) ) " { x } ) C_ ran ( u i^i ( A X. A ) )
29		rnin 5542	. . . . . . . . . . . . . . 15 |- ran ( u i^i ( A X. A ) ) C_ ( ran u i^i ran ( A X. A ) )
30	28, 29	sstri 3612	. . . . . . . . . . . . . 14 |- ( ( u i^i ( A X. A ) ) " { x } ) C_ ( ran u i^i ran ( A X. A ) )
31		inss2 3834	. . . . . . . . . . . . . 14 |- ( ran u i^i ran ( A X. A ) ) C_ ran ( A X. A )
32	30, 31	sstri 3612	. . . . . . . . . . . . 13 |- ( ( u i^i ( A X. A ) ) " { x } ) C_ ran ( A X. A )
33		rnxpid 5567	. . . . . . . . . . . . 13 |- ran ( A X. A ) = A
34	32, 33	sseqtri 3637	. . . . . . . . . . . 12 |- ( ( u i^i ( A X. A ) ) " { x } ) C_ A
35	34	a1i 11	. . . . . . . . . . 11 |- ( ( u " { x } ) C_ a -> ( ( u i^i ( A X. A ) ) " { x } ) C_ A )
36	27, 35	ssind 3837	. . . . . . . . . 10 |- ( ( u " { x } ) C_ a -> ( ( u i^i ( A X. A ) ) " { x } ) C_ ( a i^i A ) )
37	36	adantl 482	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) /\ u e. U ) /\ ( u " { x } ) C_ a ) -> ( ( u i^i ( A X. A ) ) " { x } ) C_ ( a i^i A ) )
38		simpllr 799	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) /\ u e. U ) /\ ( u " { x } ) C_ a ) -> b = ( a i^i A ) )
39	37, 38	sseqtr4d 3642	. . . . . . . 8 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) /\ u e. U ) /\ ( u " { x } ) C_ a ) -> ( ( u i^i ( A X. A ) ) " { x } ) C_ b )
40		imaeq1 5461	. . . . . . . . . 10 |- ( v = ( u i^i ( A X. A ) ) -> ( v " { x } ) = ( ( u i^i ( A X. A ) ) " { x } ) )
41	40	sseq1d 3632	. . . . . . . . 9 |- ( v = ( u i^i ( A X. A ) ) -> ( ( v " { x } ) C_ b <-> ( ( u i^i ( A X. A ) ) " { x } ) C_ b ) )
42	41	rspcev 3309	. . . . . . . 8 |- ( ( ( u i^i ( A X. A ) ) e. ( U ↾t ( A X. A ) ) /\ ( ( u i^i ( A X. A ) ) " { x } ) C_ b ) -> E. v e. ( U ↾t ( A X. A ) ) ( v " { x } ) C_ b )
43	22, 39, 42	syl2anc 693	. . . . . . 7 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) /\ u e. U ) /\ ( u " { x } ) C_ a ) -> E. v e. ( U ↾t ( A X. A ) ) ( v " { x } ) C_ b )
44		simplr 792	. . . . . . . 8 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) -> a e. (unifTop ` U ) )
45		inss1 3833	. . . . . . . . 9 |- ( a i^i A ) C_ a
46		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) -> x e. b )
47		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) -> b = ( a i^i A ) )
48	46, 47	eleqtrd 2703	. . . . . . . . 9 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) -> x e. ( a i^i A ) )
49	45, 48	sseldi 3601	. . . . . . . 8 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) -> x e. a )
50		elutop 22037	. . . . . . . . . 10 |- ( U e. (UnifOn ` X ) -> ( a e. (unifTop ` U ) <-> ( a C_ X /\ A. x e. a E. u e. U ( u " { x } ) C_ a ) ) )
51	50	simplbda 654	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ a e. (unifTop ` U ) ) -> A. x e. a E. u e. U ( u " { x } ) C_ a )
52	51	r19.21bi 2932	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ a e. (unifTop ` U ) ) /\ x e. a ) -> E. u e. U ( u " { x } ) C_ a )
53	15, 44, 49, 52	syl21anc 1325	. . . . . . 7 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) -> E. u e. U ( u " { x } ) C_ a )
54	43, 53	r19.29a 3078	. . . . . 6 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) /\ a e. (unifTop ` U ) ) /\ b = ( a i^i A ) ) -> E. v e. ( U ↾t ( A X. A ) ) ( v " { x } ) C_ b )
55	9	adantr 481	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) -> E. a e. (unifTop ` U ) b = ( a i^i A ) )
56	54, 55	r19.29a 3078	. . . . 5 |- ( ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) /\ x e. b ) -> E. v e. ( U ↾t ( A X. A ) ) ( v " { x } ) C_ b )
57	56	ralrimiva 2966	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) -> A. x e. b E. v e. ( U ↾t ( A X. A ) ) ( v " { x } ) C_ b )
58		trust 22033	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
59		elutop 22037	. . . . . 6 |- ( ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) -> ( b e. (unifTop ` ( U ↾t ( A X. A ) ) ) <-> ( b C_ A /\ A. x e. b E. v e. ( U ↾t ( A X. A ) ) ( v " { x } ) C_ b ) ) )
60	58, 59	syl 17	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( b e. (unifTop ` ( U ↾t ( A X. A ) ) ) <-> ( b C_ A /\ A. x e. b E. v e. ( U ↾t ( A X. A ) ) ( v " { x } ) C_ b ) ) )
61	60	biimpar 502	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ ( b C_ A /\ A. x e. b E. v e. ( U ↾t ( A X. A ) ) ( v " { x } ) C_ b ) ) -> b e. (unifTop ` ( U ↾t ( A X. A ) ) ) )
62	1, 14, 57, 61	syl12anc 1324	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) -> b e. (unifTop ` ( U ↾t ( A X. A ) ) ) )
63	62	ex 450	. 2 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( b e. ( (unifTop ` U ) ↾t A ) -> b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) )
64	63	ssrdv 3609	1 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( (unifTop ` U ) ↾t A ) C_ (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   ran crn 5115   "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:  restutopopn  22042