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Theorem metuel 20151
Description: Elementhood in the uniform structure generated by a metric  D (Contributed by Thierry Arnoux, 8-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
metuel  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( V  e.  (metUnif `  D )  <->  ( V  C_  ( X  X.  X )  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V
) ) )
Distinct variable groups:    w, a, D    X, a    w, V
Allowed substitution hints:    V( a)    X( w)

Proof of Theorem metuel
Dummy variable  e is distinct from all other variables.
StepHypRef Expression
1 metuval 20123 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) )
21adantl 466 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) )
32eleq2d 2508 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( V  e.  (metUnif `  D )  <->  V  e.  ( ( X  X.  X ) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) ) ) )
4 oveq2 6097 . . . . . . 7  |-  ( a  =  e  ->  (
0 [,) a )  =  ( 0 [,) e ) )
54imaeq2d 5167 . . . . . 6  |-  ( a  =  e  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) e
) ) )
65cbvmptv 4381 . . . . 5  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
76rneqi 5064 . . . 4  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
87metustfbas 20139 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
9 elfg 19442 . . 3  |-  ( ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  e.  ( fBas `  ( X  X.  X
) )  ->  ( V  e.  ( ( X  X.  X ) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )  <->  ( V  C_  ( X  X.  X
)  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) w 
C_  V ) ) )
108, 9syl 16 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( V  e.  ( ( X  X.  X ) filGen ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )  <->  ( V  C_  ( X  X.  X
)  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) w 
C_  V ) ) )
113, 10bitrd 253 1  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( V  e.  (metUnif `  D )  <->  ( V  C_  ( X  X.  X )  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   E.wrex 2714    C_ wss 3326   (/)c0 3635    e. cmpt 4348    X. cxp 4836   `'ccnv 4837   ran crn 4839   "cima 4841   ` cfv 5416  (class class class)co 6089   0cc0 9280   RR+crp 10989   [,)cico 11300  PsMetcpsmet 17798   fBascfbas 17802   filGencfg 17803  metUnifcmetu 17806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-rp 10990  df-ico 11304  df-psmet 17807  df-fbas 17812  df-fg 17813  df-metu 17815
This theorem is referenced by:  metuel2  20152  metustbl  20154  restmetu  20160
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