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Theorem metustbl 21593
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 1009 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  D  e.  (PsMet `  X ) )
2 simp3 1011 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  P  e.  X )
3 simpr 463 . . . . 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 3050 . . . . . . . 8  |-  w  e. 
_V
5 eqid 2453 . . . . . . . . 9  |-  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  =  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )
65elrnmpt 5084 . . . . . . . 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 198 . . . . . 6  |-  ( w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  ->  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r ) ) )
98ad2antlr 734 . . . . 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 3455 . . . . . . 7  |-  ( w  =  ( `' D " ( 0 [,) r
) )  ->  (
w  C_  V  <->  ( `' D " ( 0 [,) r ) )  C_  V ) )
1110biimpcd 228 . . . . . 6  |-  ( w 
C_  V  ->  (
w  =  ( `' D " ( 0 [,) r ) )  ->  ( `' D " ( 0 [,) r
) )  C_  V
) )
1211reximdv 2863 . . . . 5  |-  ( w 
C_  V  ->  ( E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r
) )  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r
) )  C_  V
) )
133, 9, 12sylc 62 . . . 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
)
14 ne0i 3739 . . . . . 6  |-  ( P  e.  X  ->  X  =/=  (/) )
152, 14syl 17 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  X  =/=  (/) )
16 simp2 1010 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  V  e.  (metUnif `  D ) )
17 metuel 21591 . . . . . 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
) ) )
1817simplbda 630 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  V  e.  (metUnif `  D ) )  ->  E. w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) ) w  C_  V
)
1915, 1, 16, 18syl21anc 1268 . . . 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 )
2013, 19r19.29a 2934 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r
) )  C_  V
)
21 imass1 5206 . . . . . 6  |-  ( ( `' D " ( 0 [,) r ) ) 
C_  V  ->  (
( `' D "
( 0 [,) r
) ) " { P } )  C_  ( V " { P }
) )
2221reximi 2857 . . . . 5  |-  ( E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V  ->  E. r  e.  RR+  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) )
23 blval2 21589 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  ( P ( ball `  D
) r )  =  ( ( `' D " ( 0 [,) r
) ) " { P } ) )
2423sseq1d 3461 . . . . . . 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 } ) ) )
25243expa 1209 . . . . . 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 } ) ) )
2625rexbidva 2900 . . . . 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 } ) ) )
2722, 26syl5ibr 225 . . . 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 }
) ) )
2827imp 431 . . 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 } ) )
291, 2, 20, 28syl21anc 1268 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) )
30 blssexps 21453 . . 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 }
) ) )
31303adant2 1028 . 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 }
) ) )
3229, 31mpbird 236 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 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   E.wrex 2740   _Vcvv 3047    C_ wss 3406   (/)c0 3733   {csn 3970    |-> cmpt 4464    X. cxp 4835   `'ccnv 4836   ran crn 4838   "cima 4840   ` cfv 5585  (class class class)co 6295   0cc0 9544   RR+crp 11309   [,)cico 11644  PsMetcpsmet 18966   ballcbl 18969  metUnifcmetu 18973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7961  df-inf 7962  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ico 11648  df-psmet 18974  df-bl 18977  df-fbas 18979  df-fg 18980  df-metu 18981
This theorem is referenced by:  psmetutop  21594
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