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Theorem xmeterval 21225
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 21122 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
2 ffn 5713 . . 3  |-  ( D : ( X  X.  X ) --> RR*  ->  D  Fn  ( X  X.  X ) )
3 elpreima 5984 . . 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 4400 . . 3  |-  ( A  .~  B  <->  A ( `' D " RR ) B )
7 df-br 4395 . . 3  |-  ( A ( `' D " RR ) B  <->  <. A ,  B >.  e.  ( `' D " RR ) )
86, 7bitri 249 . 2  |-  ( A  .~  B  <->  <. A ,  B >.  e.  ( `' D " RR ) )
9 df-3an 976 . . 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 4852 . . . . 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 2479 . . . 4  |-  ( ( A D B )  e.  RR  <->  ( D `  <. A ,  B >. )  e.  RR )
1411, 13anbi12i 695 . . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   <.cop 3977   class class class wbr 4394    X. cxp 4820   `'ccnv 4821   "cima 4825    Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277   RRcr 9520   RR*cxr 9656   *Metcxmt 18721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458  df-xr 9661  df-xmet 18730
This theorem is referenced by:  xmeter  21226  xmetec  21227  xmetresbl  21230  xrsblre  21606  isbndx  31540
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