Theorem iscfilu 22092

Description: The predicate " F is a Cauchy filter base on uniform space U." (Contributed by Thierry Arnoux, 18-Nov-2017.)

Assertion

Ref	Expression
iscfilu	|- ( U e. (UnifOn ` X ) -> ( F e. (CauFilu ` U ) <-> ( F e. ( fBas ` X ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) ) )

Distinct variable groups:   v, a, F   v, U

Allowed substitution hints:   U( a)   X( v, a)

Proof of Theorem iscfilu

Dummy variables f u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrnust 22028	. . . . 5 |- ( U e. (UnifOn ` X ) -> U e. U. ran UnifOn )
2		unieq 4444	. . . . . . . . 9 |- ( u = U -> U. u = U. U )
3	2	dmeqd 5326	. . . . . . . 8 |- ( u = U -> dom U. u = dom U. U )
4	3	fveq2d 6195	. . . . . . 7 |- ( u = U -> ( fBas ` dom U. u ) = ( fBas ` dom U. U ) )
5		raleq 3138	. . . . . . 7 |- ( u = U -> ( A. v e. u E. a e. f ( a X. a ) C_ v <-> A. v e. U E. a e. f ( a X. a ) C_ v ) )
6	4, 5	rabeqbidv 3195	. . . . . 6 |- ( u = U -> { f e. ( fBas ` dom U. u ) | A. v e. u E. a e. f ( a X. a ) C_ v } = { f e. ( fBas ` dom U. U ) | A. v e. U E. a e. f ( a X. a ) C_ v } )
7		df-cfilu 22091	. . . . . 6 |- CauFilu = ( u e. U. ran UnifOn |-> { f e. ( fBas ` dom U. u ) | A. v e. u E. a e. f ( a X. a ) C_ v } )
8		fvex 6201	. . . . . . 7 |- ( fBas ` dom U. U ) e. _V
9	8	rabex 4813	. . . . . 6 |- { f e. ( fBas ` dom U. U ) | A. v e. U E. a e. f ( a X. a ) C_ v } e. _V
10	6, 7, 9	fvmpt 6282	. . . . 5 |- ( U e. U. ran UnifOn -> (CauFilu ` U ) = { f e. ( fBas ` dom U. U ) | A. v e. U E. a e. f ( a X. a ) C_ v } )
11	1, 10	syl 17	. . . 4 |- ( U e. (UnifOn ` X ) -> (CauFilu ` U ) = { f e. ( fBas ` dom U. U ) | A. v e. U E. a e. f ( a X. a ) C_ v } )
12	11	eleq2d 2687	. . 3 |- ( U e. (UnifOn ` X ) -> ( F e. (CauFilu ` U ) <-> F e. { f e. ( fBas ` dom U. U ) | A. v e. U E. a e. f ( a X. a ) C_ v } ) )
13		rexeq 3139	. . . . 5 |- ( f = F -> ( E. a e. f ( a X. a ) C_ v <-> E. a e. F ( a X. a ) C_ v ) )
14	13	ralbidv 2986	. . . 4 |- ( f = F -> ( A. v e. U E. a e. f ( a X. a ) C_ v <-> A. v e. U E. a e. F ( a X. a ) C_ v ) )
15	14	elrab 3363	. . 3 |- ( F e. { f e. ( fBas ` dom U. U ) | A. v e. U E. a e. f ( a X. a ) C_ v } <-> ( F e. ( fBas ` dom U. U ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) )
16	12, 15	syl6bb 276	. 2 |- ( U e. (UnifOn ` X ) -> ( F e. (CauFilu ` U ) <-> ( F e. ( fBas ` dom U. U ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) ) )
17		ustbas2 22029	. . . . 5 |- ( U e. (UnifOn ` X ) -> X = dom U. U )
18	17	fveq2d 6195	. . . 4 |- ( U e. (UnifOn ` X ) -> ( fBas ` X ) = ( fBas ` dom U. U ) )
19	18	eleq2d 2687	. . 3 |- ( U e. (UnifOn ` X ) -> ( F e. ( fBas ` X ) <-> F e. ( fBas ` dom U. U ) ) )
20	19	anbi1d 741	. 2 |- ( U e. (UnifOn ` X ) -> ( ( F e. ( fBas ` X ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) <-> ( F e. ( fBas ` dom U. U ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) ) )
21	16, 20	bitr4d 271	1 |- ( U e. (UnifOn ` X ) -> ( F e. (CauFilu ` U ) <-> ( F e. ( fBas ` X ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   U.cuni 4436   X. cxp 5112   dom cdm 5114   ran crn 5115   ` cfv 5888   fBascfbas 19734  UnifOncust 22003  CauFil_uccfilu 22090

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-cfilu 22091

This theorem is referenced by:  cfilufbas  22093  cfiluexsm  22094  fmucnd  22096  cfilufg  22097  trcfilu  22098  cfiluweak  22099  neipcfilu  22100  cfilucfil  22364