Theorem xmeterval 13075

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

Step	Hyp	Ref	Expression
1		xmetf 12990	. . 3 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
2		ffn 5337	. . 3 |- ( D : ( X X. X ) --> RR* -> D Fn ( X X. X ) )
3		elpreima 5604	. . 3 |- ( D Fn ( X X. X ) -> ( <. A , B >. e. ( `' D " RR ) <-> ( <. A , B >. e. ( X X. X ) /\ ( D ` <. A , B >. ) e. RR ) ) )
4	1, 2, 3	3syl 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 )
6	5	breqi 3988	. . 3 |- ( A .~ B <-> A ( `' D " RR ) B )
7		df-br 3983	. . 3 |- ( A ( `' D " RR ) B <-> <. A , B >. e. ( `' D " RR ) )
8	6, 7	bitri 183	. 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 4634	. . . . 5 |- ( <. A , B >. e. ( X X. X ) <-> ( A e. X /\ B e. X ) )
11	10	bicomi 131	. . . 4 |- ( ( A e. X /\ B e. X ) <-> <. A , B >. e. ( X X. X ) )
12		df-ov 5845	. . . . 5 |- ( A D B ) = ( D ` <. A , B >. )
13	12	eleq1i 2232	. . . 4 |- ( ( A D B ) e. RR <-> ( D ` <. A , B >. ) e. RR )
14	11, 13	anbi12i 456	. . 3 |- ( ( ( A e. X /\ B e. X ) /\ ( A D B ) e. RR ) <-> ( <. A , B >. e. ( X X. X ) /\ ( D ` <. A , B >. ) e. RR ) )
15	9, 14	bitri 183	. 2 |- ( ( A e. X /\ B e. X /\ ( A D B ) e. RR ) <-> ( <. A , B >. e. ( X X. X ) /\ ( D ` <. A , B >. ) e. RR ) )
16	4, 8, 15	3bitr4g 222	1 |- ( D e. ( *Met ` X ) -> ( A .~ B <-> ( A e. X /\ B e. X /\ ( A D B ) e. RR ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   <.cop 3579   class class class wbr 3982   X. cxp 4602   `'ccnv 4603   "cima 4607   Fn wfn 5183   -->wf 5184   ` cfv 5188  (class class class)co 5842   RRcr 7752   RR*cxr 7932   *Metcxmet 12620

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-pnf 7935  df-mnf 7936  df-xr 7937  df-xmet 12628

This theorem is referenced by:  xmeter  13076  xmetec  13077  xmetresbl  13080