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Theorem xmeterval 13595
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 13510 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
2 ffn 5361 . . 3  |-  ( D : ( X  X.  X ) --> RR*  ->  D  Fn  ( X  X.  X ) )
3 elpreima 5631 . . 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 4006 . . 3  |-  ( A  .~  B  <->  A ( `' D " RR ) B )
7 df-br 4001 . . 3  |-  ( A ( `' D " RR ) B  <->  <. A ,  B >.  e.  ( `' D " RR ) )
86, 7bitri 184 . 2  |-  ( A  .~  B  <->  <. A ,  B >.  e.  ( `' D " RR ) )
9 df-3an 980 . . 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 4653 . . . . 5  |-  ( <. A ,  B >.  e.  ( X  X.  X
)  <->  ( A  e.  X  /\  B  e.  X ) )
1110bicomi 132 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  <->  <. A ,  B >.  e.  ( X  X.  X
) )
12 df-ov 5872 . . . . 5  |-  ( A D B )  =  ( D `  <. A ,  B >. )
1312eleq1i 2243 . . . 4  |-  ( ( A D B )  e.  RR  <->  ( D `  <. A ,  B >. )  e.  RR )
1411, 13anbi12i 460 . . 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 184 . 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 223 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   <.cop 3594   class class class wbr 4000    X. cxp 4621   `'ccnv 4622   "cima 4626    Fn wfn 5207   -->wf 5208   ` cfv 5212  (class class class)co 5869   RRcr 7798   RR*cxr 7978   *Metcxmet 13140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-pnf 7981  df-mnf 7982  df-xr 7983  df-xmet 13148
This theorem is referenced by:  xmeter  13596  xmetec  13597  xmetresbl  13600
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