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Theorem metustid 20793
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 5294 . . 3  |-  Rel  (  _I  |`  X )
21a1i 11 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  Rel  (  _I  |`  X ) )
3 vex 3111 . . . . . . . . . . . . . . 15  |-  q  e. 
_V
43brres 5273 . . . . . . . . . . . . . 14  |-  ( p (  _I  |`  X ) q  <->  ( p  _I  q  /\  p  e.  X ) )
5 df-br 4443 . . . . . . . . . . . . . 14  |-  ( p (  _I  |`  X ) q  <->  <. p ,  q
>.  e.  (  _I  |`  X ) )
63ideq 5148 . . . . . . . . . . . . . . 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 6280 . . . . . . . . . . 11  |-  ( p D p )  =  ( D `  <. p ,  p >. )
13 opeq2 4209 . . . . . . . . . . . 12  |-  ( p  =  q  ->  <. p ,  p >.  =  <. p ,  q >. )
1413fveq2d 5863 . . . . . . . . . . 11  |-  ( p  =  q  ->  ( D `  <. p ,  p >. )  =  ( D `  <. p ,  q >. )
)
1512, 14syl5eq 2515 . . . . . . . . . 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 757 . . . . . . . . . 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 20542 . . . . . . . . . 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 2505 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( D `  <. p ,  q >.
)  =  0 )
22 0xr 9631 . . . . . . . . . . 11  |-  0  e.  RR*
2322a1i 11 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  0  e. 
RR* )
24 rpxr 11218 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  a  e. 
RR* )
25 rpgt0 11222 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  0  < 
a )
26 lbico1 11570 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  a  e.  RR*  /\  0  < 
a )  ->  0  e.  ( 0 [,) a
) )
2723, 24, 25, 26syl3anc 1223 . . . . . . . . 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 2550 . . . . . . 7  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( D `  <. p ,  q >.
)  e.  ( 0 [,) a ) )
30 psmetf 20540 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
31 ffun 5726 . . . . . . . . . 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 2551 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  q  e.  X
)
35 opelxpi 5025 . . . . . . . . . 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 5728 . . . . . . . . . . 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 2553 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  <. p ,  q
>.  e.  dom  D )
41 fvimacnv 5989 . . . . . . . 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 2553 . . . 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 754 . . . . 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 20785 . . . . . 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 2998 . . 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 5088 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 1374    e. wcel 1762   E.wrex 2810    C_ wss 3471   <.cop 4028   class class class wbr 4442    |-> cmpt 4500    _I cid 4785    X. cxp 4992   `'ccnv 4993   dom cdm 4994   ran crn 4995    |` cres 4996   "cima 4997   Rel wrel 4999   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6277   0cc0 9483   RR*cxr 9618    < clt 9619   RR+crp 11211   [,)cico 11522  PsMetcpsmet 18168
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-i2m1 9551  ax-1ne0 9552  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558
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-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  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-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-rp 11212  df-ico 11526  df-psmet 18177
This theorem is referenced by:  metustfbas  20799  metust  20801
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