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Theorem xmeterval 20665
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 20562 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
2 ffn 5724 . . 3  |-  ( D : ( X  X.  X ) --> RR*  ->  D  Fn  ( X  X.  X ) )
3 elpreima 5994 . . 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 20 . 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 4448 . . 3  |-  ( A  .~  B  <->  A ( `' D " RR ) B )
7 df-br 4443 . . 3  |-  ( A ( `' D " RR ) B  <->  <. A ,  B >.  e.  ( `' D " RR ) )
86, 7bitri 249 . 2  |-  ( A  .~  B  <->  <. A ,  B >.  e.  ( `' D " RR ) )
9 df-3an 970 . . 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 5023 . . . . 5  |-  ( <. A ,  B >.  e.  ( X  X.  X
)  <->  ( A  e.  X  /\  B  e.  X ) )
1110bicomi 202 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  <->  <. A ,  B >.  e.  ( X  X.  X
) )
12 df-ov 6280 . . . . 5  |-  ( A D B )  =  ( D `  <. A ,  B >. )
1312eleq1i 2539 . . . 4  |-  ( ( A D B )  e.  RR  <->  ( D `  <. A ,  B >. )  e.  RR )
1411, 13anbi12i 697 . . 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 249 . 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 288 1  |-  ( D  e.  ( *Met `  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( A D B )  e.  RR ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   <.cop 4028   class class class wbr 4442    X. cxp 4992   `'ccnv 4993   "cima 4997    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   RRcr 9482   RR*cxr 9618   *Metcxmt 18169
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  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-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-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-xr 9623  df-xmet 18178
This theorem is referenced by:  xmeter  20666  xmetec  20667  xmetresbl  20670  xrsblre  21046  isbndx  29870
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