Theorem restutop 18220

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 444	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) -> ( U e. (UnifOn ` X ) /\ A C_ X ) )
2		fvex 5701	. . . . . . . 8 |- (unifTop ` U ) e. _V
3	2	a1i 11	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> (unifTop ` U ) e. _V )
4		elfvex 5717	. . . . . . . . 9 |- ( U e. (UnifOn ` X ) -> X e. _V )
5	4	adantr 452	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> X e. _V )
6		simpr 448	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> A C_ X )
7	5, 6	ssexd 4310	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> A e. _V )
8		elrest 13610	. . . . . . 7 |- ( ( (unifTop ` U ) e. _V /\ A e. _V ) -> ( b e. ( (unifTop ` U ) ↾t A ) <-> E. a e. (unifTop ` U ) b = ( a i^i A ) ) )
9	3, 7, 8	syl2anc 643	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( b e. ( (unifTop ` U ) ↾t A ) <-> E. a e. (unifTop ` U ) b = ( a i^i A ) ) )
10	9	biimpa 471	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) -> E. a e. (unifTop ` U ) b = ( a i^i A ) )
11		inss2 3522	. . . . . . 7 |- ( a i^i A ) C_ A
12		sseq1 3329	. . . . . . 7 |- ( b = ( a i^i A ) -> ( b C_ A <-> ( a i^i A ) C_ A ) )
13	11, 12	mpbiri 225	. . . . . 6 |- ( b = ( a i^i A ) -> b C_ A )
14	13	rexlimivw 2786	. . . . 5 |- ( E. a e. (unifTop ` U ) b = ( a i^i A ) -> b C_ A )
15	10, 14	syl 16	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) -> b C_ A )
16		simp-5l 745	. . . . . . . . . 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 ) )
17	16	ad2antrr 707	. . . . . . . . 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 ) )
18	7	ad6antr 717	. . . . . . . . . 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 )
19		xpexg 4948	. . . . . . . . . 10 |- ( ( A e. _V /\ A e. _V ) -> ( A X. A ) e. _V )
20	18, 18, 19	syl2anc 643	. . . . . . . . 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 )
21		simplr 732	. . . . . . . . 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 )
22		elrestr 13611	. . . . . . . . 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 ) ) )
23	17, 20, 21, 22	syl3anc 1184	. . . . . . . 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 ) ) )
24		inss1 3521	. . . . . . . . . . . . 13 |- ( u i^i ( A X. A ) ) C_ u
25		imass1 5198	. . . . . . . . . . . . 13 |- ( ( u i^i ( A X. A ) ) C_ u -> ( ( u i^i ( A X. A ) ) " { x } ) C_ ( u " { x } ) )
26	24, 25	ax-mp 8	. . . . . . . . . . . 12 |- ( ( u i^i ( A X. A ) ) " { x } ) C_ ( u " { x } )
27		sstr 3316	. . . . . . . . . . . 12 |- ( ( ( ( u i^i ( A X. A ) ) " { x } ) C_ ( u " { x } ) /\ ( u " { x } ) C_ a ) -> ( ( u i^i ( A X. A ) ) " { x } ) C_ a )
28	26, 27	mpan 652	. . . . . . . . . . 11 |- ( ( u " { x } ) C_ a -> ( ( u i^i ( A X. A ) ) " { x } ) C_ a )
29		imassrn 5175	. . . . . . . . . . . . . . 15 |- ( ( u i^i ( A X. A ) ) " { x } ) C_ ran ( u i^i ( A X. A ) )
30		rnin 5240	. . . . . . . . . . . . . . 15 |- ran ( u i^i ( A X. A ) ) C_ ( ran u i^i ran ( A X. A ) )
31	29, 30	sstri 3317	. . . . . . . . . . . . . 14 |- ( ( u i^i ( A X. A ) ) " { x } ) C_ ( ran u i^i ran ( A X. A ) )
32		inss2 3522	. . . . . . . . . . . . . 14 |- ( ran u i^i ran ( A X. A ) ) C_ ran ( A X. A )
33	31, 32	sstri 3317	. . . . . . . . . . . . 13 |- ( ( u i^i ( A X. A ) ) " { x } ) C_ ran ( A X. A )
34		rnxpid 5261	. . . . . . . . . . . . 13 |- ran ( A X. A ) = A
35	33, 34	sseqtri 3340	. . . . . . . . . . . 12 |- ( ( u i^i ( A X. A ) ) " { x } ) C_ A
36	35	a1i 11	. . . . . . . . . . 11 |- ( ( u " { x } ) C_ a -> ( ( u i^i ( A X. A ) ) " { x } ) C_ A )
37	28, 36	ssind 3525	. . . . . . . . . 10 |- ( ( u " { x } ) C_ a -> ( ( u i^i ( A X. A ) ) " { x } ) C_ ( a i^i A ) )
38	37	adantl 453	. . . . . . . . 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 ) )
39		simpllr 736	. . . . . . . . 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 ) )
40	38, 39	sseqtr4d 3345	. . . . . . . 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 )
41		imaeq1 5157	. . . . . . . . . 10 |- ( v = ( u i^i ( A X. A ) ) -> ( v " { x } ) = ( ( u i^i ( A X. A ) ) " { x } ) )
42	41	sseq1d 3335	. . . . . . . . 9 |- ( v = ( u i^i ( A X. A ) ) -> ( ( v " { x } ) C_ b <-> ( ( u i^i ( A X. A ) ) " { x } ) C_ b ) )
43	42	rspcev 3012	. . . . . . . 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 )
44	23, 40, 43	syl2anc 643	. . . . . . 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 )
45		simplr 732	. . . . . . . 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 ) )
46		inss1 3521	. . . . . . . . 9 |- ( a i^i A ) C_ a
47		simpllr 736	. . . . . . . . . 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 )
48		simpr 448	. . . . . . . . . 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 ) )
49	47, 48	eleqtrd 2480	. . . . . . . . 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 ) )
50	46, 49	sseldi 3306	. . . . . . . 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 )
51		elutop 18216	. . . . . . . . . 10 |- ( U e. (UnifOn ` X ) -> ( a e. (unifTop ` U ) <-> ( a C_ X /\ A. x e. a E. u e. U ( u " { x } ) C_ a ) ) )
52	51	simplbda 608	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ a e. (unifTop ` U ) ) -> A. x e. a E. u e. U ( u " { x } ) C_ a )
53	52	r19.21bi 2764	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ a e. (unifTop ` U ) ) /\ x e. a ) -> E. u e. U ( u " { x } ) C_ a )
54	16, 45, 50, 53	syl21anc 1183	. . . . . . 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 )
55	44, 54	r19.29a 2810	. . . . . 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 )
56	10	adantr 452	. . . . . 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 ) )
57	55, 56	r19.29a 2810	. . . . 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 )
58	57	ralrimiva 2749	. . . 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 )
59		trust 18212	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
60		elutop 18216	. . . . . 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 ) ) )
61	59, 60	syl 16	. . . . 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 ) ) )
62	61	biimpar 472	. . . 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 ) ) ) )
63	1, 15, 58, 62	syl12anc 1182	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X ) /\ b e. ( (unifTop ` U ) ↾t A ) ) -> b e. (unifTop ` ( U ↾t ( A X. A ) ) ) )
64	63	ex 424	. 2 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( b e. ( (unifTop ` U ) ↾t A ) -> b e. (unifTop ` ( U ↾t ( A X. A ) ) ) ) )
65	64	ssrdv 3314	1 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( (unifTop ` U ) ↾t A ) C_ (unifTop ` ( U ↾t ( A X. A ) ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916   i^i cin 3279   C_ wss 3280   {csn 3774   X. cxp 4835   ran crn 4838   "cima 4840   ` cfv 5413  (class class class)co 6040   ↾_t crest 13603  UnifOncust 18182  unifTopcutop 18213

This theorem is referenced by:  restutopopn  18221

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-rest 13605  df-ust 18183  df-utop 18214