Theorem utopbas 22039

Description: The base of the topology induced by a uniform structure U. (Contributed by Thierry Arnoux, 5-Dec-2017.)

Assertion

Ref	Expression
utopbas	|- ( U e. (UnifOn ` X ) -> X = U. (unifTop ` U ) )

Proof of Theorem utopbas

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

Step	Hyp	Ref	Expression
1		utopval 22036	. . . 4 |- ( U e. (UnifOn ` X ) -> (unifTop ` U ) = { a e. ~P X | A. x e. a E. v e. U ( v " { x } ) C_ a } )
2		ssrab2 3687	. . . 4 |- { a e. ~P X | A. x e. a E. v e. U ( v " { x } ) C_ a } C_ ~P X
3	1, 2	syl6eqss 3655	. . 3 |- ( U e. (UnifOn ` X ) -> (unifTop ` U ) C_ ~P X )
4		ssid 3624	. . . . . 6 |- X C_ X
5	4	a1i 11	. . . . 5 |- ( U e. (UnifOn ` X ) -> X C_ X )
6		ustssxp 22008	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ v e. U ) -> v C_ ( X X. X ) )
7		imassrn 5477	. . . . . . . . . 10 |- ( v " { x } ) C_ ran v
8		rnss 5354	. . . . . . . . . . 11 |- ( v C_ ( X X. X ) -> ran v C_ ran ( X X. X ) )
9		rnxpid 5567	. . . . . . . . . . 11 |- ran ( X X. X ) = X
10	8, 9	syl6sseq 3651	. . . . . . . . . 10 |- ( v C_ ( X X. X ) -> ran v C_ X )
11	7, 10	syl5ss 3614	. . . . . . . . 9 |- ( v C_ ( X X. X ) -> ( v " { x } ) C_ X )
12	6, 11	syl 17	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ v e. U ) -> ( v " { x } ) C_ X )
13	12	ralrimiva 2966	. . . . . . 7 |- ( U e. (UnifOn ` X ) -> A. v e. U ( v " { x } ) C_ X )
14		ustne0 22017	. . . . . . . 8 |- ( U e. (UnifOn ` X ) -> U =/= (/) )
15		r19.2zb 4061	. . . . . . . 8 |- ( U =/= (/) <-> ( A. v e. U ( v " { x } ) C_ X -> E. v e. U ( v " { x } ) C_ X ) )
16	14, 15	sylib 208	. . . . . . 7 |- ( U e. (UnifOn ` X ) -> ( A. v e. U ( v " { x } ) C_ X -> E. v e. U ( v " { x } ) C_ X ) )
17	13, 16	mpd 15	. . . . . 6 |- ( U e. (UnifOn ` X ) -> E. v e. U ( v " { x } ) C_ X )
18	17	ralrimivw 2967	. . . . 5 |- ( U e. (UnifOn ` X ) -> A. x e. X E. v e. U ( v " { x } ) C_ X )
19		elutop 22037	. . . . 5 |- ( U e. (UnifOn ` X ) -> ( X e. (unifTop ` U ) <-> ( X C_ X /\ A. x e. X E. v e. U ( v " { x } ) C_ X ) ) )
20	5, 18, 19	mpbir2and 957	. . . 4 |- ( U e. (UnifOn ` X ) -> X e. (unifTop ` U ) )
21		elpwuni 4616	. . . 4 |- ( X e. (unifTop ` U ) -> ( (unifTop ` U ) C_ ~P X <-> U. (unifTop ` U ) = X ) )
22	20, 21	syl 17	. . 3 |- ( U e. (UnifOn ` X ) -> ( (unifTop ` U ) C_ ~P X <-> U. (unifTop ` U ) = X ) )
23	3, 22	mpbid 222	. 2 |- ( U e. (UnifOn ` X ) -> U. (unifTop ` U ) = X )
24	23	eqcomd 2628	1 |- ( U e. (UnifOn ` X ) -> X = U. (unifTop ` U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   X. cxp 5112   ran crn 5115   "cima 5117   ` cfv 5888  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-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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ust 22004  df-utop 22035

This theorem is referenced by:  utoptopon  22040  utop2nei  22054  utopreg  22056  tuslem  22071