Theorem elutop 22037

Description: Open sets in the topology induced by an uniform structure U on X (Contributed by Thierry Arnoux, 30-Nov-2017.)

Assertion

Ref	Expression
elutop	|- ( U e. (UnifOn ` X ) -> ( A e. (unifTop ` U ) <-> ( A C_ X /\ A. x e. A E. v e. U ( v " { x } ) C_ A ) ) )

Distinct variable groups:   x, v, A   v, U, x   x, X

Allowed substitution hint:   X( v)

Proof of Theorem elutop

Dummy variable a is 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	1	eleq2d 2687	. . 3 |- ( U e. (UnifOn ` X ) -> ( A e. (unifTop ` U ) <-> A e. { a e. ~P X | A. x e. a E. v e. U ( v " { x } ) C_ a } ) )
3		sseq2 3627	. . . . . 6 |- ( a = A -> ( ( v " { x } ) C_ a <-> ( v " { x } ) C_ A ) )
4	3	rexbidv 3052	. . . . 5 |- ( a = A -> ( E. v e. U ( v " { x } ) C_ a <-> E. v e. U ( v " { x } ) C_ A ) )
5	4	raleqbi1dv 3146	. . . 4 |- ( a = A -> ( A. x e. a E. v e. U ( v " { x } ) C_ a <-> A. x e. A E. v e. U ( v " { x } ) C_ A ) )
6	5	elrab 3363	. . 3 |- ( A e. { a e. ~P X | A. x e. a E. v e. U ( v " { x } ) C_ a } <-> ( A e. ~P X /\ A. x e. A E. v e. U ( v " { x } ) C_ A ) )
7	2, 6	syl6bb 276	. 2 |- ( U e. (UnifOn ` X ) -> ( A e. (unifTop ` U ) <-> ( A e. ~P X /\ A. x e. A E. v e. U ( v " { x } ) C_ A ) ) )
8		elex 3212	. . . . 5 |- ( A e. ~P X -> A e. _V )
9	8	a1i 11	. . . 4 |- ( U e. (UnifOn ` X ) -> ( A e. ~P X -> A e. _V ) )
10		elfvex 6221	. . . . . . 7 |- ( U e. (UnifOn ` X ) -> X e. _V )
11	10	adantr 481	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> X e. _V )
12		simpr 477	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> A C_ X )
13	11, 12	ssexd 4805	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> A e. _V )
14	13	ex 450	. . . 4 |- ( U e. (UnifOn ` X ) -> ( A C_ X -> A e. _V ) )
15		elpwg 4166	. . . . 5 |- ( A e. _V -> ( A e. ~P X <-> A C_ X ) )
16	15	a1i 11	. . . 4 |- ( U e. (UnifOn ` X ) -> ( A e. _V -> ( A e. ~P X <-> A C_ X ) ) )
17	9, 14, 16	pm5.21ndd 369	. . 3 |- ( U e. (UnifOn ` X ) -> ( A e. ~P X <-> A C_ X ) )
18	17	anbi1d 741	. 2 |- ( U e. (UnifOn ` X ) -> ( ( A e. ~P X /\ A. x e. A E. v e. U ( v " { x } ) C_ A ) <-> ( A C_ X /\ A. x e. A E. v e. U ( v " { x } ) C_ A ) ) )
19	7, 18	bitrd 268	1 |- ( U e. (UnifOn ` X ) -> ( A e. (unifTop ` U ) <-> ( A C_ X /\ A. x e. A E. v e. U ( v " { x } ) C_ A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   "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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ust 22004  df-utop 22035

This theorem is referenced by:  utoptop  22038  utopbas  22039  restutop  22041  restutopopn  22042  ucncn  22089