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Theorem xmeterval 12641
Description: Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
Hypothesis
Ref Expression
xmeter.1  |-  .~  =  ( `' D " RR )
Assertion
Ref Expression
xmeterval  |-  ( D  e.  ( *Met `  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( A D B )  e.  RR ) ) )

Proof of Theorem xmeterval
StepHypRef Expression
1 xmetf 12556 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
2 ffn 5279 . . 3  |-  ( D : ( X  X.  X ) --> RR*  ->  D  Fn  ( X  X.  X ) )
3 elpreima 5546 . . 3  |-  ( D  Fn  ( X  X.  X )  ->  ( <. A ,  B >.  e.  ( `' D " RR )  <->  ( <. A ,  B >.  e.  ( X  X.  X )  /\  ( D `  <. A ,  B >. )  e.  RR ) ) )
41, 2, 33syl 17 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( <. A ,  B >.  e.  ( `' D " RR )  <->  ( <. A ,  B >.  e.  ( X  X.  X )  /\  ( D `  <. A ,  B >. )  e.  RR ) ) )
5 xmeter.1 . . . 4  |-  .~  =  ( `' D " RR )
65breqi 3942 . . 3  |-  ( A  .~  B  <->  A ( `' D " RR ) B )
7 df-br 3937 . . 3  |-  ( A ( `' D " RR ) B  <->  <. A ,  B >.  e.  ( `' D " RR ) )
86, 7bitri 183 . 2  |-  ( A  .~  B  <->  <. A ,  B >.  e.  ( `' D " RR ) )
9 df-3an 965 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  ( A D B )  e.  RR )  <->  ( ( A  e.  X  /\  B  e.  X )  /\  ( A D B )  e.  RR ) )
10 opelxp 4576 . . . . 5  |-  ( <. A ,  B >.  e.  ( X  X.  X
)  <->  ( A  e.  X  /\  B  e.  X ) )
1110bicomi 131 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  <->  <. A ,  B >.  e.  ( X  X.  X
) )
12 df-ov 5784 . . . . 5  |-  ( A D B )  =  ( D `  <. A ,  B >. )
1312eleq1i 2206 . . . 4  |-  ( ( A D B )  e.  RR  <->  ( D `  <. A ,  B >. )  e.  RR )
1411, 13anbi12i 456 . . 3  |-  ( ( ( A  e.  X  /\  B  e.  X
)  /\  ( A D B )  e.  RR ) 
<->  ( <. A ,  B >.  e.  ( X  X.  X )  /\  ( D `  <. A ,  B >. )  e.  RR ) )
159, 14bitri 183 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  ( A D B )  e.  RR )  <->  ( <. A ,  B >.  e.  ( X  X.  X )  /\  ( D `  <. A ,  B >. )  e.  RR ) )
164, 8, 153bitr4g 222 1  |-  ( D  e.  ( *Met `  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( A D B )  e.  RR ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332    e. wcel 1481   <.cop 3534   class class class wbr 3936    X. cxp 4544   `'ccnv 4545   "cima 4549    Fn wfn 5125   -->wf 5126   ` cfv 5130  (class class class)co 5781   RRcr 7642   RR*cxr 7822   *Metcxmet 12186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-map 6551  df-pnf 7825  df-mnf 7826  df-xr 7827  df-xmet 12194
This theorem is referenced by:  xmeter  12642  xmetec  12643  xmetresbl  12646
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