Theorem cfilucfil3OLD 20964

Description: Given a metric D and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.)

Assertion

Ref	Expression
cfilucfil3OLD	|- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( ( C e. ( Fil ` X ) /\ C e. (CauFilu ` (metUnifOLD ` D ) ) ) <-> C e. (CauFil ` D ) ) )

Proof of Theorem cfilucfil3OLD

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfilucfil2OLD 20283	. . . 4 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( C e. (CauFilu ` (metUnifOLD ` D ) ) <-> ( C e. ( fBas ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
2	1	anbi2d 703	. . 3 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( ( C e. ( Fil ` X ) /\ C e. (CauFilu ` (metUnifOLD ` D ) ) ) <-> ( C e. ( Fil ` X ) /\ ( C e. ( fBas ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) ) )
3		filfbas 19556	. . . . . 6 |- ( C e. ( Fil ` X ) -> C e. ( fBas ` X ) )
4	3	pm4.71i 632	. . . . 5 |- ( C e. ( Fil ` X ) <-> ( C e. ( Fil ` X ) /\ C e. ( fBas ` X ) ) )
5	4	anbi1i 695	. . . 4 |- ( ( C e. ( Fil ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) <-> ( ( C e. ( Fil ` X ) /\ C e. ( fBas ` X ) ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) )
6		anass 649	. . . 4 |- ( ( ( C e. ( Fil ` X ) /\ C e. ( fBas ` X ) ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) <-> ( C e. ( Fil ` X ) /\ ( C e. ( fBas ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
7	5, 6	bitr2i 250	. . 3 |- ( ( C e. ( Fil ` X ) /\ ( C e. ( fBas ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) <-> ( C e. ( Fil ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) )
8	2, 7	syl6bb 261	. 2 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( ( C e. ( Fil ` X ) /\ C e. (CauFilu ` (metUnifOLD ` D ) ) ) <-> ( C e. ( Fil ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
9		iscfil 20911	. . 3 |- ( D e. ( *Met ` X ) -> ( C e. (CauFil ` D ) <-> ( C e. ( Fil ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
10	9	adantl 466	. 2 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( C e. (CauFil ` D ) <-> ( C e. ( Fil ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
11	8, 10	bitr4d 256	1 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( ( C e. ( Fil ` X ) /\ C e. (CauFilu ` (metUnifOLD ` D ) ) ) <-> C e. (CauFil ` D ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1758   =/= wne 2648   A.wral 2799   E.wrex 2800   C_ wss 3439   (/)c0 3748   X. cxp 4949   "cima 4954   ` cfv 5529  (class class class)co 6203   0cc0 9396   RR+crp 11105   [,)cico 11416   *Metcxmt 17929   fBascfbas 17932  metUnif_OLDcmetuOLD 17935   Filcfil 19553  CauFil_uccfilu 19996  CauFilccfil 20898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-2 10494  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-ico 11420  df-xmet 17938  df-fbas 17942  df-fg 17943  df-metuOLD 17944  df-fil 19554  df-ust 19910  df-cfilu 19997  df-cfil 20901

This theorem is referenced by:  cfilucfil4OLD  20966