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Theorem metuel 21206
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 21178 . . . 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 2527 . 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 6304 . . . . . . 7  |-  ( a  =  e  ->  (
0 [,) a )  =  ( 0 [,) e ) )
54imaeq2d 5347 . . . . . 6  |-  ( a  =  e  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) e
) ) )
65cbvmptv 4548 . . . . 5  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
76rneqi 5239 . . . 4  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
87metustfbas 21194 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
9 elfg 20497 . . 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 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    C_ wss 3471   (/)c0 3793    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   ran crn 5009   "cima 5011   ` cfv 5594  (class class class)co 6296   0cc0 9509   RR+crp 11245   [,)cico 11556  PsMetcpsmet 18528   fBascfbas 18532   filGencfg 18533  metUnifcmetu 18536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-rp 11246  df-ico 11560  df-psmet 18537  df-fbas 18542  df-fg 18543  df-metu 18545
This theorem is referenced by:  metuel2  21207  metustbl  21209  restmetu  21215
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