Theorem xmeterval 14755

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 14670	. . 3 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
2		ffn 5410	. . 3 |- ( D : ( X X. X ) --> RR* -> D Fn ( X X. X ) )
3		elpreima 5684	. . 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 4040	. . 3 |- ( A .~ B <-> A ( `' D " RR ) B )
7		df-br 4035	. . 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 982	. . 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 4694	. . . . 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 5928	. . . . 5 |- ( A D B ) = ( D ` <. A , B >. )
13	12	eleq1i 2262	. . . 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 980   = wceq 1364   e. wcel 2167   <.cop 3626   class class class wbr 4034   X. cxp 4662   `'ccnv 4663   "cima 4667   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5925   RRcr 7895   RR*cxr 8077   *Metcxmet 14168

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-xmet 14176

This theorem is referenced by:  xmeter  14756  xmetec  14757  xmetresbl  14760