Theorem metres 7823

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 1475	. . . 4 |- dom dom D = dom dom D
2	1	metf 7807	. . 3 |- (D e. Met -> D:(dom dom D X. dom dom D)-->RR)
3		fdm 3631	. . 3 |- (D:(dom dom D X. dom dom D)-->RR -> dom D = (dom dom D X. dom dom D))
4		ineq2 2211	. . . . . 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 3269	. . . . . 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 1523	. . . . 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 3369	. . . . 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 10	. . . 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 3497	. . . . 5 |- (D |` dom ( D |` (R X. R))) = (D |` (R X. R))
10		dmres 3380	. . . . . 6 |- dom ( D |` (R X. R)) = ((R X. R) i^i dom D)
11		reseq2 3369	. . . . . 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	eqtr3 1497	. . . 4 |- (D |` (R X. R)) = (D |` ((R X. R) i^i dom D))
14	8, 13	syl5eq 1519	. . 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 20	. 2 |- (D e. Met -> (D |` (R X. R)) = (D |` ((R i^i dom dom D) X. (R i^i dom dom D))))
16		inss2 2231	. . 3 |- (R i^i dom dom D) (_ dom dom D
17		eqid 1475	. . . 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 7822	. . 3 |- ((D e. Met /\ (R i^i dom dom D) (_ dom dom D) -> (D |` ((R i^i dom dom D) X. (R i^i dom dom D))) e. Met)
19	16, 18	mpan2 696	. 2 |- (D e. Met -> (D |` ((R i^i dom dom D) X. (R i^i dom dom D))) e. Met)
20	15, 19	eqeltrd 1548	1 |- (D e. Met -> (D |` (R X. R)) e. Met)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958   i^i cin 2046   (_ wss 2047   X. cxp 3168  dom cdm 3170   |` cres 3172  -->wf 3178  RRcr 5233  Metcme 7789

This theorem is referenced by:  cncfmet 7905  remet 7910  lmsslem 7952  lmss 7953  caussi 7954  causs 7955  cmsss 7997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-met 7793

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