Theorem isusp 22065

Description: The predicate W is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)

Hypotheses

Ref	Expression
isusp.1	|- B = ( Base ` W )
isusp.2	|- U = (UnifSt ` W )
isusp.3	|- J = ( TopOpen ` W )

Assertion

Ref	Expression
isusp	|- ( W e. UnifSp <-> ( U e. (UnifOn ` B ) /\ J = (unifTop ` U ) ) )

Proof of Theorem isusp

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( W e. UnifSp -> W e. _V )
2		0nep0 4836	. . . . 5 |- (/) =/= { (/) }
3		isusp.1	. . . . . . . . . . . 12 |- B = ( Base ` W )
4		fvprc 6185	. . . . . . . . . . . 12 |- ( -. W e. _V -> ( Base ` W ) = (/) )
5	3, 4	syl5eq 2668	. . . . . . . . . . 11 |- ( -. W e. _V -> B = (/) )
6	5	fveq2d 6195	. . . . . . . . . 10 |- ( -. W e. _V -> (UnifOn ` B ) = (UnifOn ` (/) ) )
7		ust0 22023	. . . . . . . . . 10 |- (UnifOn ` (/) ) = { { (/) } }
8	6, 7	syl6eq 2672	. . . . . . . . 9 |- ( -. W e. _V -> (UnifOn ` B ) = { { (/) } } )
9	8	eleq2d 2687	. . . . . . . 8 |- ( -. W e. _V -> ( U e. (UnifOn ` B ) <-> U e. { { (/) } } ) )
10		isusp.2	. . . . . . . . . 10 |- U = (UnifSt ` W )
11		fvex 6201	. . . . . . . . . 10 |- (UnifSt ` W ) e. _V
12	10, 11	eqeltri 2697	. . . . . . . . 9 |- U e. _V
13	12	elsn 4192	. . . . . . . 8 |- ( U e. { { (/) } } <-> U = { (/) } )
14	9, 13	syl6bb 276	. . . . . . 7 |- ( -. W e. _V -> ( U e. (UnifOn ` B ) <-> U = { (/) } ) )
15		fvprc 6185	. . . . . . . . 9 |- ( -. W e. _V -> (UnifSt ` W ) = (/) )
16	10, 15	syl5eq 2668	. . . . . . . 8 |- ( -. W e. _V -> U = (/) )
17	16	eqeq1d 2624	. . . . . . 7 |- ( -. W e. _V -> ( U = { (/) } <-> (/) = { (/) } ) )
18	14, 17	bitrd 268	. . . . . 6 |- ( -. W e. _V -> ( U e. (UnifOn ` B ) <-> (/) = { (/) } ) )
19	18	necon3bbid 2831	. . . . 5 |- ( -. W e. _V -> ( -. U e. (UnifOn ` B ) <-> (/) =/= { (/) } ) )
20	2, 19	mpbiri 248	. . . 4 |- ( -. W e. _V -> -. U e. (UnifOn ` B ) )
21	20	con4i 113	. . 3 |- ( U e. (UnifOn ` B ) -> W e. _V )
22	21	adantr 481	. 2 |- ( ( U e. (UnifOn ` B ) /\ J = (unifTop ` U ) ) -> W e. _V )
23		fveq2 6191	. . . . . 6 |- ( w = W -> (UnifSt ` w ) = (UnifSt ` W ) )
24	23, 10	syl6eqr 2674	. . . . 5 |- ( w = W -> (UnifSt ` w ) = U )
25		fveq2 6191	. . . . . . 7 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
26	25, 3	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( Base ` w ) = B )
27	26	fveq2d 6195	. . . . 5 |- ( w = W -> (UnifOn ` ( Base ` w ) ) = (UnifOn ` B ) )
28	24, 27	eleq12d 2695	. . . 4 |- ( w = W -> ( (UnifSt ` w ) e. (UnifOn ` ( Base ` w ) ) <-> U e. (UnifOn ` B ) ) )
29		fveq2 6191	. . . . . 6 |- ( w = W -> ( TopOpen ` w ) = ( TopOpen ` W ) )
30		isusp.3	. . . . . 6 |- J = ( TopOpen ` W )
31	29, 30	syl6eqr 2674	. . . . 5 |- ( w = W -> ( TopOpen ` w ) = J )
32	24	fveq2d 6195	. . . . 5 |- ( w = W -> (unifTop ` (UnifSt ` w ) ) = (unifTop ` U ) )
33	31, 32	eqeq12d 2637	. . . 4 |- ( w = W -> ( ( TopOpen ` w ) = (unifTop ` (UnifSt ` w ) ) <-> J = (unifTop ` U ) ) )
34	28, 33	anbi12d 747	. . 3 |- ( w = W -> ( ( (UnifSt ` w ) e. (UnifOn ` ( Base ` w ) ) /\ ( TopOpen ` w ) = (unifTop ` (UnifSt ` w ) ) ) <-> ( U e. (UnifOn ` B ) /\ J = (unifTop ` U ) ) ) )
35		df-usp 22061	. . 3 |- UnifSp = { w | ( (UnifSt ` w ) e. (UnifOn ` ( Base ` w ) ) /\ ( TopOpen ` w ) = (unifTop ` (UnifSt ` w ) ) ) }
36	34, 35	elab2g 3353	. 2 |- ( W e. _V -> ( W e. UnifSp <-> ( U e. (UnifOn ` B ) /\ J = (unifTop ` U ) ) ) )
37	1, 22, 36	pm5.21nii 368	1 |- ( W e. UnifSp <-> ( U e. (UnifOn ` B ) /\ J = (unifTop ` U ) ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   {csn 4177   ` cfv 5888   Basecbs 15857   TopOpenctopn 16082  UnifOncust 22003  unifTopcutop 22034  UnifStcuss 22057  UnifSpcusp 22058

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  df-usp 22061

This theorem is referenced by:  ressust  22068  ressusp  22069  tususp  22076  uspreg  22078  ucncn  22089  neipcfilu  22100  ucnextcn  22108  xmsusp  22374