Theorem ust0 22023

Description: The unique uniform structure of the empty set is the empty set. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)

Assertion

Ref	Expression
ust0	|- (UnifOn ` (/) ) = { { (/) } }

Proof of Theorem ust0

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4790	. . . . . . . 8 |- (/) e. _V
2		isust 22007	. . . . . . . 8 |- ( (/) e. _V -> ( u e. (UnifOn ` (/) ) <-> ( u C_ ~P ( (/) X. (/) ) /\ ( (/) X. (/) ) e. u /\ A. v e. u ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` (/) ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) ) )
3	1, 2	ax-mp 5	. . . . . . 7 |- ( u e. (UnifOn ` (/) ) <-> ( u C_ ~P ( (/) X. (/) ) /\ ( (/) X. (/) ) e. u /\ A. v e. u ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` (/) ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) )
4	3	simp1bi 1076	. . . . . 6 |- ( u e. (UnifOn ` (/) ) -> u C_ ~P ( (/) X. (/) ) )
5		0xp 5199	. . . . . . . 8 |- ( (/) X. (/) ) = (/)
6	5	pweqi 4162	. . . . . . 7 |- ~P ( (/) X. (/) ) = ~P (/)
7		pw0 4343	. . . . . . 7 |- ~P (/) = { (/) }
8	6, 7	eqtri 2644	. . . . . 6 |- ~P ( (/) X. (/) ) = { (/) }
9	4, 8	syl6sseq 3651	. . . . 5 |- ( u e. (UnifOn ` (/) ) -> u C_ { (/) } )
10		ustbasel 22010	. . . . . . 7 |- ( u e. (UnifOn ` (/) ) -> ( (/) X. (/) ) e. u )
11	5, 10	syl5eqelr 2706	. . . . . 6 |- ( u e. (UnifOn ` (/) ) -> (/) e. u )
12	11	snssd 4340	. . . . 5 |- ( u e. (UnifOn ` (/) ) -> { (/) } C_ u )
13	9, 12	eqssd 3620	. . . 4 |- ( u e. (UnifOn ` (/) ) -> u = { (/) } )
14		velsn 4193	. . . 4 |- ( u e. { { (/) } } <-> u = { (/) } )
15	13, 14	sylibr 224	. . 3 |- ( u e. (UnifOn ` (/) ) -> u e. { { (/) } } )
16	15	ssriv 3607	. 2 |- (UnifOn ` (/) ) C_ { { (/) } }
17	8	eqimss2i 3660	. . . 4 |- { (/) } C_ ~P ( (/) X. (/) )
18	1	snid 4208	. . . . 5 |- (/) e. { (/) }
19	5, 18	eqeltri 2697	. . . 4 |- ( (/) X. (/) ) e. { (/) }
20	18	a1i 11	. . . . . 6 |- ( (/) C_ (/) -> (/) e. { (/) } )
21	8	raleqi 3142	. . . . . . 7 |- ( A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) <-> A. w e. { (/) } ( (/) C_ w -> w e. { (/) } ) )
22		sseq2 3627	. . . . . . . . 9 |- ( w = (/) -> ( (/) C_ w <-> (/) C_ (/) ) )
23		eleq1 2689	. . . . . . . . 9 |- ( w = (/) -> ( w e. { (/) } <-> (/) e. { (/) } ) )
24	22, 23	imbi12d 334	. . . . . . . 8 |- ( w = (/) -> ( ( (/) C_ w -> w e. { (/) } ) <-> ( (/) C_ (/) -> (/) e. { (/) } ) ) )
25	1, 24	ralsn 4222	. . . . . . 7 |- ( A. w e. { (/) } ( (/) C_ w -> w e. { (/) } ) <-> ( (/) C_ (/) -> (/) e. { (/) } ) )
26	21, 25	bitri 264	. . . . . 6 |- ( A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) <-> ( (/) C_ (/) -> (/) e. { (/) } ) )
27	20, 26	mpbir 221	. . . . 5 |- A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } )
28		inidm 3822	. . . . . . 7 |- ( (/) i^i (/) ) = (/)
29	28, 18	eqeltri 2697	. . . . . 6 |- ( (/) i^i (/) ) e. { (/) }
30		ineq2 3808	. . . . . . . 8 |- ( w = (/) -> ( (/) i^i w ) = ( (/) i^i (/) ) )
31	30	eleq1d 2686	. . . . . . 7 |- ( w = (/) -> ( ( (/) i^i w ) e. { (/) } <-> ( (/) i^i (/) ) e. { (/) } ) )
32	1, 31	ralsn 4222	. . . . . 6 |- ( A. w e. { (/) } ( (/) i^i w ) e. { (/) } <-> ( (/) i^i (/) ) e. { (/) } )
33	29, 32	mpbir 221	. . . . 5 |- A. w e. { (/) } ( (/) i^i w ) e. { (/) }
34		res0 5400	. . . . . . 7 |- ( _I |` (/) ) = (/)
35	34	eqimssi 3659	. . . . . 6 |- ( _I |` (/) ) C_ (/)
36		cnv0 5535	. . . . . . 7 |- `' (/) = (/)
37	36, 18	eqeltri 2697	. . . . . 6 |- `' (/) e. { (/) }
38		0trrel 13720	. . . . . . 7 |- ( (/) o. (/) ) C_ (/)
39		id 22	. . . . . . . . . 10 |- ( w = (/) -> w = (/) )
40	39, 39	coeq12d 5286	. . . . . . . . 9 |- ( w = (/) -> ( w o. w ) = ( (/) o. (/) ) )
41	40	sseq1d 3632	. . . . . . . 8 |- ( w = (/) -> ( ( w o. w ) C_ (/) <-> ( (/) o. (/) ) C_ (/) ) )
42	1, 41	rexsn 4223	. . . . . . 7 |- ( E. w e. { (/) } ( w o. w ) C_ (/) <-> ( (/) o. (/) ) C_ (/) )
43	38, 42	mpbir 221	. . . . . 6 |- E. w e. { (/) } ( w o. w ) C_ (/)
44	35, 37, 43	3pm3.2i 1239	. . . . 5 |- ( ( _I |` (/) ) C_ (/) /\ `' (/) e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ (/) )
45		sseq1 3626	. . . . . . . . 9 |- ( v = (/) -> ( v C_ w <-> (/) C_ w ) )
46	45	imbi1d 331	. . . . . . . 8 |- ( v = (/) -> ( ( v C_ w -> w e. { (/) } ) <-> ( (/) C_ w -> w e. { (/) } ) ) )
47	46	ralbidv 2986	. . . . . . 7 |- ( v = (/) -> ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) <-> A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) ) )
48		ineq1 3807	. . . . . . . . 9 |- ( v = (/) -> ( v i^i w ) = ( (/) i^i w ) )
49	48	eleq1d 2686	. . . . . . . 8 |- ( v = (/) -> ( ( v i^i w ) e. { (/) } <-> ( (/) i^i w ) e. { (/) } ) )
50	49	ralbidv 2986	. . . . . . 7 |- ( v = (/) -> ( A. w e. { (/) } ( v i^i w ) e. { (/) } <-> A. w e. { (/) } ( (/) i^i w ) e. { (/) } ) )
51		sseq2 3627	. . . . . . . 8 |- ( v = (/) -> ( ( _I |` (/) ) C_ v <-> ( _I |` (/) ) C_ (/) ) )
52		cnveq 5296	. . . . . . . . 9 |- ( v = (/) -> `' v = `' (/) )
53	52	eleq1d 2686	. . . . . . . 8 |- ( v = (/) -> ( `' v e. { (/) } <-> `' (/) e. { (/) } ) )
54		sseq2 3627	. . . . . . . . 9 |- ( v = (/) -> ( ( w o. w ) C_ v <-> ( w o. w ) C_ (/) ) )
55	54	rexbidv 3052	. . . . . . . 8 |- ( v = (/) -> ( E. w e. { (/) } ( w o. w ) C_ v <-> E. w e. { (/) } ( w o. w ) C_ (/) ) )
56	51, 53, 55	3anbi123d 1399	. . . . . . 7 |- ( v = (/) -> ( ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) <-> ( ( _I |` (/) ) C_ (/) /\ `' (/) e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ (/) ) ) )
57	47, 50, 56	3anbi123d 1399	. . . . . 6 |- ( v = (/) -> ( ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( v i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) ) <-> ( A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( (/) i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ (/) /\ `' (/) e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ (/) ) ) ) )
58	1, 57	ralsn 4222	. . . . 5 |- ( A. v e. { (/) } ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( v i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) ) <-> ( A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( (/) i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ (/) /\ `' (/) e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ (/) ) ) )
59	27, 33, 44, 58	mpbir3an 1244	. . . 4 |- A. v e. { (/) } ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( v i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) )
60		isust 22007	. . . . 5 |- ( (/) e. _V -> ( { (/) } e. (UnifOn ` (/) ) <-> ( { (/) } C_ ~P ( (/) X. (/) ) /\ ( (/) X. (/) ) e. { (/) } /\ A. v e. { (/) } ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( v i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) ) ) ) )
61	1, 60	ax-mp 5	. . . 4 |- ( { (/) } e. (UnifOn ` (/) ) <-> ( { (/) } C_ ~P ( (/) X. (/) ) /\ ( (/) X. (/) ) e. { (/) } /\ A. v e. { (/) } ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( v i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) ) ) )
62	17, 19, 59, 61	mpbir3an 1244	. . 3 |- { (/) } e. (UnifOn ` (/) )
63		snssi 4339	. . 3 |- ( { (/) } e. (UnifOn ` (/) ) -> { { (/) } } C_ (UnifOn ` (/) ) )
64	62, 63	ax-mp 5	. 2 |- { { (/) } } C_ (UnifOn ` (/) )
65	16, 64	eqssi 3619	1 |- (UnifOn ` (/) ) = { { (/) } }

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888  UnifOncust 22003

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-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-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-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-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ust 22004

This theorem is referenced by:  isusp  22065