Theorem metres 9100

Description: A restriction of a metric is a metric.

Assertion

Ref	Expression
metres	|- (D e. Met -> (D |` (R X. R)) e. Met)

Proof of Theorem metres

Step	Hyp	Ref	Expression
1		eqid 1884	. . . 4 |- dom dom D = dom dom D
2	1	metf 9084	. . 3 |- (D e. Met -> D:(dom dom D X. dom dom D)-->RR)
3		fdm 4567	. . 3 |- (D:(dom dom D X. dom dom D)-->RR -> dom D = (dom dom D X. dom dom D))
4		ineq2 2790	. . . . . 6 |- (dom D = (dom dom D X. dom dom D) -> ((R X. R) i^i dom D) = ((R X. R) i^i (dom dom D X. dom dom D)))
5		inxp 4109	. . . . . 6 |- ((R X. R) i^i (dom dom D X. dom dom D)) = ((R i^i dom dom D) X. (R i^i dom dom D))
6	4, 5	syl6eq 1944	. . . . 5 |- (dom D = (dom dom D X. dom dom D) -> ((R X. R) i^i dom D) = ((R i^i dom dom D) X. (R i^i dom dom D)))
7		reseq2 4219	. . . . 5 |- (((R X. R) i^i dom D) = ((R i^i dom dom D) X. (R i^i dom dom D)) -> (D |` ((R X. R) i^i dom D)) = (D |` ((R i^i dom dom D) X. (R i^i dom dom D))))
8	6, 7	syl 12	. . . 4 |- (dom D = (dom dom D X. dom dom D) -> (D |` ((R X. R) i^i dom D)) = (D |` ((R i^i dom dom D) X. (R i^i dom dom D))))
9		resdmres 4390	. . . . 5 |- (D |` dom ( D |` (R X. R))) = (D |` (R X. R))
10		dmres 4234	. . . . . 6 |- dom ( D |` (R X. R)) = ((R X. R) i^i dom D)
11		reseq2 4219	. . . . . 6 |- (dom ( D |` (R X. R)) = ((R X. R) i^i dom D) -> (D |` dom ( D |` (R X. R))) = (D |` ((R X. R) i^i dom D)))
12	10, 11	ax-mp 7	. . . . 5 |- (D |` dom ( D |` (R X. R))) = (D |` ((R X. R) i^i dom D))
13	9, 12	eqtr3i 1910	. . . 4 |- (D |` (R X. R)) = (D |` ((R X. R) i^i dom D))
14	8, 13	syl5eq 1940	. . 3 |- (dom D = (dom dom D X. dom dom D) -> (D |` (R X. R)) = (D |` ((R i^i dom dom D) X. (R i^i dom dom D))))
15	2, 3, 14	3syl 24	. 2 |- (D e. Met -> (D |` (R X. R)) = (D |` ((R i^i dom dom D) X. (R i^i dom dom D))))
16		inss2 2813	. . 3 |- (R i^i dom dom D) C_ dom dom D
17		eqid 1884	. . . 4 |- dom dom ( D |` ((R i^i dom dom D) X. (R i^i dom dom D))) = dom dom ( D |` ((R i^i dom dom D) X. (R i^i dom dom D)))
18	1, 17	metreslem 9099	. . 3 |- ((D e. Met /\ (R i^i dom dom D) C_ dom dom D) -> (D |` ((R i^i dom dom D) X. (R i^i dom dom D))) e. Met)
19	16, 18	mpan2 760	. 2 |- (D e. Met -> (D |` ((R i^i dom dom D) X. (R i^i dom dom D))) e. Met)
20	15, 19	eqeltrd 1971	1 |- (D e. Met -> (D |` (R X. R)) e. Met)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300   i^i cin 2592   C_ wss 2593   X. cxp 3984  dom cdm 3986   |` cres 3988  -->wf 3994  RRcr 6385  Metcme 9066

This theorem is referenced by:  cncfmet 9183  remet 9188  lmsslem 9230  lmss 9231  caussi 9232  causs 9233  cmsss 9275  subtopmetlem 10255  subtopmet 10256  blssp 15844  metdcn 15853  iitop 15867  iiuni 15868  cncfco 15887  sstotbnd 15936  totbndss 15937  bndss 15942  blbnd 15943  ismtyres 15954  heiborlem23 15977  rrntotbnd 16022  rrnheibor 16023  reheibor 16025  iccbnd 16026  phtpycolem3 16053  phtpycolem4 16054  pcocn 16076  pcohtpylem3 16082

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-met 9070

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