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Theorem metustid 21647
Description: The identity diagonal is included in all elements of the filter base generated by the metric  D. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
Assertion
Ref Expression
metustid  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  (  _I  |`  X )  C_  A )
Distinct variable groups:    D, a    X, a    A, a    F, a

Proof of Theorem metustid
Dummy variables  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5138 . . 3  |-  Rel  (  _I  |`  X )
21a1i 11 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  Rel  (  _I  |`  X ) )
3 vex 3034 . . . . . . . . . . . . . . 15  |-  q  e. 
_V
43brres 5117 . . . . . . . . . . . . . 14  |-  ( p (  _I  |`  X ) q  <->  ( p  _I  q  /\  p  e.  X ) )
5 df-br 4396 . . . . . . . . . . . . . 14  |-  ( p (  _I  |`  X ) q  <->  <. p ,  q
>.  e.  (  _I  |`  X ) )
63ideq 4992 . . . . . . . . . . . . . . 15  |-  ( p  _I  q  <->  p  =  q )
76anbi1i 709 . . . . . . . . . . . . . 14  |-  ( ( p  _I  q  /\  p  e.  X )  <->  ( p  =  q  /\  p  e.  X )
)
84, 5, 73bitr3i 283 . . . . . . . . . . . . 13  |-  ( <.
p ,  q >.  e.  (  _I  |`  X )  <-> 
( p  =  q  /\  p  e.  X
) )
98biimpi 199 . . . . . . . . . . . 12  |-  ( <.
p ,  q >.  e.  (  _I  |`  X )  ->  ( p  =  q  /\  p  e.  X ) )
109ad2antlr 741 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( p  =  q  /\  p  e.  X ) )
1110simpld 466 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  p  =  q )
12 df-ov 6311 . . . . . . . . . . 11  |-  ( p D p )  =  ( D `  <. p ,  p >. )
13 opeq2 4159 . . . . . . . . . . . 12  |-  ( p  =  q  ->  <. p ,  p >.  =  <. p ,  q >. )
1413fveq2d 5883 . . . . . . . . . . 11  |-  ( p  =  q  ->  ( D `  <. p ,  p >. )  =  ( D `  <. p ,  q >. )
)
1512, 14syl5eq 2517 . . . . . . . . . 10  |-  ( p  =  q  ->  (
p D p )  =  ( D `  <. p ,  q >.
) )
1611, 15syl 17 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( p D p )  =  ( D `  <. p ,  q >. )
)
17 simplll 776 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  D  e.  (PsMet `  X ) )
1810simprd 470 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  p  e.  X
)
19 psmet0 21402 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  p  e.  X )  ->  (
p D p )  =  0 )
2017, 18, 19syl2anc 673 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( p D p )  =  0 )
2116, 20eqtr3d 2507 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( D `  <. p ,  q >.
)  =  0 )
22 0xr 9705 . . . . . . . . . . 11  |-  0  e.  RR*
2322a1i 11 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  0  e. 
RR* )
24 rpxr 11332 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  a  e. 
RR* )
25 rpgt0 11336 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  0  < 
a )
26 lbico1 11714 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  a  e.  RR*  /\  0  < 
a )  ->  0  e.  ( 0 [,) a
) )
2723, 24, 25, 26syl3anc 1292 . . . . . . . . 9  |-  ( a  e.  RR+  ->  0  e.  ( 0 [,) a
) )
2827adantl 473 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  0  e.  ( 0 [,) a ) )
2921, 28eqeltrd 2549 . . . . . . 7  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( D `  <. p ,  q >.
)  e.  ( 0 [,) a ) )
30 psmetf 21400 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
31 ffun 5742 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
3230, 31syl 17 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  Fun  D )
3332ad3antrrr 744 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  Fun  D )
3411, 18eqeltrrd 2550 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  q  e.  X
)
35 opelxpi 4871 . . . . . . . . . 10  |-  ( ( p  e.  X  /\  q  e.  X )  -> 
<. p ,  q >.  e.  ( X  X.  X
) )
3618, 34, 35syl2anc 673 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  <. p ,  q
>.  e.  ( X  X.  X ) )
37 fdm 5745 . . . . . . . . . . 11  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
3830, 37syl 17 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
3938ad3antrrr 744 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  dom  D  =  ( X  X.  X
) )
4036, 39eleqtrrd 2552 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  <. p ,  q
>.  e.  dom  D )
41 fvimacnv 6012 . . . . . . . 8  |-  ( ( Fun  D  /\  <. p ,  q >.  e.  dom  D )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
4233, 40, 41syl2anc 673 . . . . . . 7  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( ( D `
 <. p ,  q
>. )  e.  (
0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
4329, 42mpbid 215 . . . . . 6  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  <. p ,  q
>.  e.  ( `' D " ( 0 [,) a
) ) )
4443adantr 472 . . . . 5  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) )
45 simpr 468 . . . . 5  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  A  =  ( `' D " ( 0 [,) a ) ) )
4644, 45eleqtrrd 2552 . . . 4  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  <. p ,  q >.  e.  A )
47 simplr 770 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  A  e.  F )
48 metust.1 . . . . . . 7  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
4948metustel 21643 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
5049ad2antrr 740 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
5147, 50mpbid 215 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
5246, 51r19.29a 2918 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  <. p ,  q >.  e.  A
)
5352ex 441 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  ( <. p ,  q >.  e.  (  _I  |`  X )  ->  <. p ,  q
>.  e.  A ) )
542, 53relssdv 4932 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  (  _I  |`  X )  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757    C_ wss 3390   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    _I cid 4749    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Rel wrel 4844   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   RR*cxr 9692    < clt 9693   RR+crp 11325   [,)cico 11662  PsMetcpsmet 19031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-i2m1 9625  ax-1ne0 9626  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-rp 11326  df-ico 11666  df-psmet 19039
This theorem is referenced by:  metustfbas  21650  metust  21651
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