Theorem xmeterval 13974

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 13889	. . 3 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
2		ffn 5367	. . 3 |- ( D : ( X X. X ) --> RR* -> D Fn ( X X. X ) )
3		elpreima 5637	. . 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 4011	. . 3 |- ( A .~ B <-> A ( `' D " RR ) B )
7		df-br 4006	. . 3 |- ( A ( `' D " RR ) B <-> <. A , B >. e. ( `' D " RR ) )
8	6, 7	bitri 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 4658	. . . . 5 |- ( <. A , B >. e. ( X X. X ) <-> ( A e. X /\ B e. X ) )
11	10	bicomi 132	. . . 4 |- ( ( A e. X /\ B e. X ) <-> <. A , B >. e. ( X X. X ) )
12		df-ov 5880	. . . . 5 |- ( A D B ) = ( D ` <. A , B >. )
13	12	eleq1i 2243	. . . 4 |- ( ( A D B ) e. RR <-> ( D ` <. A , B >. ) e. RR )
14	11, 13	anbi12i 460	. . 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 184	. 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 223	1 |- ( D e. ( *Met ` X ) -> ( A .~ B <-> ( A e. X /\ B e. X /\ ( A D B ) e. RR ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   <.cop 3597   class class class wbr 4005   X. cxp 4626   `'ccnv 4627   "cima 4631   Fn wfn 5213   -->wf 5214   ` cfv 5218  (class class class)co 5877   RRcr 7812   RR*cxr 7993   *Metcxmet 13479

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-pnf 7996  df-mnf 7997  df-xr 7998  df-xmet 13487

This theorem is referenced by:  xmeter  13975  xmetec  13976  xmetresbl  13979