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Theorem metustid 21040
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 5291 . . 3  |-  Rel  (  _I  |`  X )
21a1i 11 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  Rel  (  _I  |`  X ) )
3 vex 3098 . . . . . . . . . . . . . . 15  |-  q  e. 
_V
43brres 5270 . . . . . . . . . . . . . 14  |-  ( p (  _I  |`  X ) q  <->  ( p  _I  q  /\  p  e.  X ) )
5 df-br 4438 . . . . . . . . . . . . . 14  |-  ( p (  _I  |`  X ) q  <->  <. p ,  q
>.  e.  (  _I  |`  X ) )
63ideq 5145 . . . . . . . . . . . . . . 15  |-  ( p  _I  q  <->  p  =  q )
76anbi1i 695 . . . . . . . . . . . . . 14  |-  ( ( p  _I  q  /\  p  e.  X )  <->  ( p  =  q  /\  p  e.  X )
)
84, 5, 73bitr3i 275 . . . . . . . . . . . . 13  |-  ( <.
p ,  q >.  e.  (  _I  |`  X )  <-> 
( p  =  q  /\  p  e.  X
) )
98biimpi 194 . . . . . . . . . . . 12  |-  ( <.
p ,  q >.  e.  (  _I  |`  X )  ->  ( p  =  q  /\  p  e.  X ) )
109ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( p  =  q  /\  p  e.  X ) )
1110simpld 459 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  p  =  q )
12 df-ov 6284 . . . . . . . . . . 11  |-  ( p D p )  =  ( D `  <. p ,  p >. )
13 opeq2 4203 . . . . . . . . . . . 12  |-  ( p  =  q  ->  <. p ,  p >.  =  <. p ,  q >. )
1413fveq2d 5860 . . . . . . . . . . 11  |-  ( p  =  q  ->  ( D `  <. p ,  p >. )  =  ( D `  <. p ,  q >. )
)
1512, 14syl5eq 2496 . . . . . . . . . 10  |-  ( p  =  q  ->  (
p D p )  =  ( D `  <. p ,  q >.
) )
1611, 15syl 16 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( p D p )  =  ( D `  <. p ,  q >. )
)
17 simplll 759 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  D  e.  (PsMet `  X ) )
1810simprd 463 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  p  e.  X
)
19 psmet0 20789 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  p  e.  X )  ->  (
p D p )  =  0 )
2017, 18, 19syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( p D p )  =  0 )
2116, 20eqtr3d 2486 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( D `  <. p ,  q >.
)  =  0 )
22 0xr 9643 . . . . . . . . . . 11  |-  0  e.  RR*
2322a1i 11 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  0  e. 
RR* )
24 rpxr 11237 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  a  e. 
RR* )
25 rpgt0 11241 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  0  < 
a )
26 lbico1 11589 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  a  e.  RR*  /\  0  < 
a )  ->  0  e.  ( 0 [,) a
) )
2723, 24, 25, 26syl3anc 1229 . . . . . . . . 9  |-  ( a  e.  RR+  ->  0  e.  ( 0 [,) a
) )
2827adantl 466 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  0  e.  ( 0 [,) a ) )
2921, 28eqeltrd 2531 . . . . . . 7  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( D `  <. p ,  q >.
)  e.  ( 0 [,) a ) )
30 psmetf 20787 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
31 ffun 5723 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
3230, 31syl 16 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  Fun  D )
3332ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  Fun  D )
3411, 18eqeltrrd 2532 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  q  e.  X
)
35 opelxpi 5021 . . . . . . . . . 10  |-  ( ( p  e.  X  /\  q  e.  X )  -> 
<. p ,  q >.  e.  ( X  X.  X
) )
3618, 34, 35syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  <. p ,  q
>.  e.  ( X  X.  X ) )
37 fdm 5725 . . . . . . . . . . 11  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
3830, 37syl 16 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
3938ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  dom  D  =  ( X  X.  X
) )
4036, 39eleqtrrd 2534 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  <. p ,  q
>.  e.  dom  D )
41 fvimacnv 5987 . . . . . . . 8  |-  ( ( Fun  D  /\  <. p ,  q >.  e.  dom  D )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
4233, 40, 41syl2anc 661 . . . . . . 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 210 . . . . . 6  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  <. p ,  q
>.  e.  ( `' D " ( 0 [,) a
) ) )
4443adantr 465 . . . . 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 461 . . . . 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 2534 . . . 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 755 . . . . 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 21032 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
5049ad2antrr 725 . . . . 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 210 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
5246, 51r19.29a 2985 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  <. p ,  q >.  e.  A
)
5352ex 434 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  ( <. p ,  q >.  e.  (  _I  |`  X )  ->  <. p ,  q
>.  e.  A ) )
542, 53relssdv 5085 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  (  _I  |`  X )  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   E.wrex 2794    C_ wss 3461   <.cop 4020   class class class wbr 4437    |-> cmpt 4495    _I cid 4780    X. cxp 4987   `'ccnv 4988   dom cdm 4989   ran crn 4990    |` cres 4991   "cima 4992   Rel wrel 4994   Fun wfun 5572   -->wf 5574   ` cfv 5578  (class class class)co 6281   0cc0 9495   RR*cxr 9630    < clt 9631   RR+crp 11230   [,)cico 11541  PsMetcpsmet 18380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-i2m1 9563  ax-1ne0 9564  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-rp 11231  df-ico 11545  df-psmet 18389
This theorem is referenced by:  metustfbas  21046  metust  21048
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