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Theorem metustbl 20814
Description: The "section" image of an entourage at a point  P always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.)
Assertion
Ref Expression
metustbl  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. a  e.  ran  ( ball `  D
) ( P  e.  a  /\  a  C_  ( V " { P } ) ) )
Distinct variable groups:    D, a    P, a    V, a    X, a

Proof of Theorem metustbl
Dummy variables  r  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 991 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  D  e.  (PsMet `  X ) )
2 simp3 993 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  P  e.  X )
3 simpr 461 . . . . 5  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X )  /\  w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r
) ) ) )  /\  w  C_  V
)  ->  w  C_  V
)
4 vex 3111 . . . . . . . 8  |-  w  e. 
_V
5 eqid 2462 . . . . . . . . 9  |-  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  =  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )
65elrnmpt 5242 . . . . . . . 8  |-  ( w  e.  _V  ->  (
w  e.  ran  (
r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  <->  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r ) ) ) )
74, 6ax-mp 5 . . . . . . 7  |-  ( w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  <->  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r ) ) )
87biimpi 194 . . . . . 6  |-  ( w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  ->  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r ) ) )
98ad2antlr 726 . . . . 5  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X )  /\  w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r
) ) ) )  /\  w  C_  V
)  ->  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r ) ) )
10 sseq1 3520 . . . . . . . 8  |-  ( w  =  ( `' D " ( 0 [,) r
) )  ->  (
w  C_  V  <->  ( `' D " ( 0 [,) r ) )  C_  V ) )
1110biimpcd 224 . . . . . . 7  |-  ( w 
C_  V  ->  (
w  =  ( `' D " ( 0 [,) r ) )  ->  ( `' D " ( 0 [,) r
) )  C_  V
) )
1211reximdv 2932 . . . . . 6  |-  ( w 
C_  V  ->  ( E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r
) )  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r
) )  C_  V
) )
1312imp 429 . . . . 5  |-  ( ( w  C_  V  /\  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r
) ) )  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V )
143, 9, 13syl2anc 661 . . . 4  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X )  /\  w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r
) ) ) )  /\  w  C_  V
)  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r
) )  C_  V
)
15 ne0i 3786 . . . . . 6  |-  ( P  e.  X  ->  X  =/=  (/) )
162, 15syl 16 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  X  =/=  (/) )
17 simp2 992 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  V  e.  (metUnif `  D ) )
18 metuel 20811 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( V  e.  (metUnif `  D )  <->  ( V  C_  ( X  X.  X )  /\  E. w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) ) w  C_  V
) ) )
1918simplbda 624 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  V  e.  (metUnif `  D ) )  ->  E. w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) ) w  C_  V
)
2016, 1, 17, 19syl21anc 1222 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r
) ) ) w 
C_  V )
2114, 20r19.29a 2998 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r
) )  C_  V
)
22 imass1 5364 . . . . . 6  |-  ( ( `' D " ( 0 [,) r ) ) 
C_  V  ->  (
( `' D "
( 0 [,) r
) ) " { P } )  C_  ( V " { P }
) )
2322reximi 2927 . . . . 5  |-  ( E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V  ->  E. r  e.  RR+  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) )
24 blval2 20808 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  ( P ( ball `  D
) r )  =  ( ( `' D " ( 0 [,) r
) ) " { P } ) )
2524sseq1d 3526 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  (
( P ( ball `  D ) r ) 
C_  ( V " { P } )  <->  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) ) )
26253expa 1191 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  r  e.  RR+ )  ->  (
( P ( ball `  D ) r ) 
C_  ( V " { P } )  <->  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) ) )
2726rexbidva 2965 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( E. r  e.  RR+  ( P ( ball `  D
) r )  C_  ( V " { P } )  <->  E. r  e.  RR+  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) ) )
2823, 27syl5ibr 221 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) ) )
2928imp 429 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V )  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  ( V " { P } ) )
301, 2, 21, 29syl21anc 1222 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) )
31 blssexps 20659 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( E. a  e.  ran  ( ball `  D )
( P  e.  a  /\  a  C_  ( V " { P }
) )  <->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) ) )
32313adant2 1010 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  ( E. a  e.  ran  ( ball `  D ) ( P  e.  a  /\  a  C_  ( V " { P } ) )  <->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) ) )
3330, 32mpbird 232 1  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. a  e.  ran  ( ball `  D
) ( P  e.  a  /\  a  C_  ( V " { P } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   E.wrex 2810   _Vcvv 3108    C_ wss 3471   (/)c0 3780   {csn 4022    |-> cmpt 4500    X. cxp 4992   `'ccnv 4993   ran crn 4995   "cima 4997   ` cfv 5581  (class class class)co 6277   0cc0 9483   RR+crp 11211   [,)cico 11522  PsMetcpsmet 18168   ballcbl 18171  metUnifcmetu 18176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ico 11526  df-psmet 18177  df-bl 18180  df-fbas 18182  df-fg 18183  df-metu 18185
This theorem is referenced by:  psmetutop  20816
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