Theorem tmsval 22286

Description: For any metric there is an associated metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
tmsval.m	|- M = { <. ( Base ` ndx ) , X >. , <. ( dist ` ndx ) , D >. }
tmsval.k	|- K = (toMetSp ` D )

Assertion

Ref	Expression
tmsval	|- ( D e. ( *Met ` X ) -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )

Proof of Theorem tmsval

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tmsval.k	. 2 |- K = (toMetSp ` D )
2		df-tms 22127	. . . 4 |- toMetSp = ( d e. U. ran *Met |-> ( { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } sSet <. (TopSet ` ndx ) , ( MetOpen ` d ) >. ) )
3	2	a1i 11	. . 3 |- ( D e. ( *Met ` X ) -> toMetSp = ( d e. U. ran *Met |-> ( { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } sSet <. (TopSet ` ndx ) , ( MetOpen ` d ) >. ) ) )
4		dmeq 5324	. . . . . . . . 9 |- ( d = D -> dom d = dom D )
5	4	dmeqd 5326	. . . . . . . 8 |- ( d = D -> dom dom d = dom dom D )
6		xmetf 22134	. . . . . . . . . . 11 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
7		fdm 6051	. . . . . . . . . . 11 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
8	6, 7	syl 17	. . . . . . . . . 10 |- ( D e. ( *Met ` X ) -> dom D = ( X X. X ) )
9	8	dmeqd 5326	. . . . . . . . 9 |- ( D e. ( *Met ` X ) -> dom dom D = dom ( X X. X ) )
10		dmxpid 5345	. . . . . . . . 9 |- dom ( X X. X ) = X
11	9, 10	syl6eq 2672	. . . . . . . 8 |- ( D e. ( *Met ` X ) -> dom dom D = X )
12	5, 11	sylan9eqr 2678	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> dom dom d = X )
13	12	opeq2d 4409	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> <. ( Base ` ndx ) , dom dom d >. = <. ( Base ` ndx ) , X >. )
14		simpr 477	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> d = D )
15	14	opeq2d 4409	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> <. ( dist ` ndx ) , d >. = <. ( dist ` ndx ) , D >. )
16	13, 15	preq12d 4276	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } = { <. ( Base ` ndx ) , X >. , <. ( dist ` ndx ) , D >. } )
17		tmsval.m	. . . . 5 |- M = { <. ( Base ` ndx ) , X >. , <. ( dist ` ndx ) , D >. }
18	16, 17	syl6eqr 2674	. . . 4 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } = M )
19	14	fveq2d 6195	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> ( MetOpen ` d ) = ( MetOpen ` D ) )
20	19	opeq2d 4409	. . . 4 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> <. (TopSet ` ndx ) , ( MetOpen ` d ) >. = <. (TopSet ` ndx ) , ( MetOpen ` D ) >. )
21	18, 20	oveq12d 6668	. . 3 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> ( { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } sSet <. (TopSet ` ndx ) , ( MetOpen ` d ) >. ) = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
22		fvssunirn 6217	. . . 4 |- ( *Met ` X ) C_ U. ran *Met
23	22	sseli 3599	. . 3 |- ( D e. ( *Met ` X ) -> D e. U. ran *Met )
24		ovexd 6680	. . 3 |- ( D e. ( *Met ` X ) -> ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) e. _V )
25	3, 21, 23, 24	fvmptd 6288	. 2 |- ( D e. ( *Met ` X ) -> (toMetSp ` D ) = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
26	1, 25	syl5eq 2668	1 |- ( D e. ( *Met ` X ) -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {cpr 4179   <.cop 4183   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   RR*cxr 10073   ndxcnx 15854   sSet csts 15855   Basecbs 15857  TopSetcts 15947   distcds 15950   *Metcxmt 19731   MetOpencmopn 19736  toMetSpctmt 22124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-xr 10078  df-xmet 19739  df-tms 22127

This theorem is referenced by:  tmslem  22287