Theorem utopval 22036

Description: The topology induced by a uniform structure U. (Contributed by Thierry Arnoux, 30-Nov-2017.)

Assertion

Ref	Expression
utopval	|- ( U e. (UnifOn ` X ) -> (unifTop ` U ) = { a e. ~P X | A. x e. a E. v e. U ( v " { x } ) C_ a } )

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

Allowed substitution hint:   X( v)

Proof of Theorem utopval

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-utop 22035	. . 3 |- unifTop = ( u e. U. ran UnifOn |-> { a e. ~P dom U. u | A. x e. a E. v e. u ( v " { x } ) C_ a } )
2	1	a1i 11	. 2 |- ( U e. (UnifOn ` X ) -> unifTop = ( u e. U. ran UnifOn |-> { a e. ~P dom U. u | A. x e. a E. v e. u ( v " { x } ) C_ a } ) )
3		simpr 477	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> u = U )
4	3	unieqd 4446	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> U. u = U. U )
5	4	dmeqd 5326	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> dom U. u = dom U. U )
6		ustbas2 22029	. . . . . 6 |- ( U e. (UnifOn ` X ) -> X = dom U. U )
7	6	adantr 481	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> X = dom U. U )
8	5, 7	eqtr4d 2659	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> dom U. u = X )
9	8	pweqd 4163	. . 3 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> ~P dom U. u = ~P X )
10	3	rexeqdv 3145	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> ( E. v e. u ( v " { x } ) C_ a <-> E. v e. U ( v " { x } ) C_ a ) )
11	10	ralbidv 2986	. . 3 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> ( A. x e. a E. v e. u ( v " { x } ) C_ a <-> A. x e. a E. v e. U ( v " { x } ) C_ a ) )
12	9, 11	rabeqbidv 3195	. 2 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> { a e. ~P dom U. u | 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 } )
13		elrnust 22028	. 2 |- ( U e. (UnifOn ` X ) -> U e. U. ran UnifOn )
14		elfvex 6221	. . 3 |- ( U e. (UnifOn ` X ) -> X e. _V )
15		pwexg 4850	. . 3 |- ( X e. _V -> ~P X e. _V )
16		rabexg 4812	. . 3 |- ( ~P X e. _V -> { a e. ~P X | A. x e. a E. v e. U ( v " { x } ) C_ a } e. _V )
17	14, 15, 16	3syl 18	. 2 |- ( U e. (UnifOn ` X ) -> { a e. ~P X | A. x e. a E. v e. U ( v " { x } ) C_ a } e. _V )
18	2, 12, 13, 17	fvmptd 6288	1 |- ( U e. (UnifOn ` X ) -> (unifTop ` U ) = { a e. ~P X | A. x e. a E. v e. U ( v " { x } ) C_ a } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   dom cdm 5114   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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ust 22004  df-utop 22035

This theorem is referenced by:  elutop  22037  utoptop  22038  utopbas  22039  utopsnneiplem  22051  psmetutop  22372