Theorem xmeterval 15536

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 15451	. . 3 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
2		ffn 5533	. . 3 |- ( D : ( X X. X ) --> RR* -> D Fn ( X X. X ) )
3		elpreima 5828	. . 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 4136	. . 3 |- ( A .~ B <-> A ( `' D " RR ) B )
7		df-br 4131	. . 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 1011	. . 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 4804	. . . . 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 6088	. . . . 5 |- ( A D B ) = ( D ` <. A , B >. )
13	12	eleq1i 2304	. . . 4 |- ( ( A D B ) e. RR <-> ( D ` <. A , B >. ) e. RR )
14	11, 13	anbi12i 464	. . 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   <.cop 3712   class class class wbr 4130   X. cxp 4772   `'ccnv 4773   "cima 4777   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   RRcr 8178   RR*cxr 8359   *Metcxmet 14873

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-pnf 8362  df-mnf 8363  df-xr 8364  df-xmet 14881

This theorem is used by:  xmeter  15537  xmetec  15538  xmetresbl  15541