Theorem metres 7820

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 1478	. . . 4 |- dom dom D = dom dom D
2	1	metf 7804	. . 3 |- (D e. Met -> D:(dom dom D X. dom dom D)-->RR)
3		fdm 3637	. . 3 |- (D:(dom dom D X. dom dom D)-->RR -> dom D = (dom dom D X. dom dom D))
4		ineq2 2214	. . . . . 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 3275	. . . . . 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 1526	. . . . 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 3375	. . . . 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 3503	. . . . 5 |- (D |` dom ( D |` (R X. R))) = (D |` (R X. R))
10		dmres 3386	. . . . . 6 |- dom ( D |` (R X. R)) = ((R X. R) i^i dom D)
11		reseq2 3375	. . . . . 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 1500	. . . 4 |- (D |` (R X. R)) = (D |` ((R X. R) i^i dom D))
14	8, 13	syl5eq 1522	. . 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 2234	. . 3 |- (R i^i dom dom D) (_ dom dom D
17		eqid 1478	. . . 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 7819	. . 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 698	. 2 |- (D e. Met -> (D |` ((R i^i dom dom D) X. (R i^i dom dom D))) e. Met)
20	15, 19	eqeltrd 1551	1 |- (D e. Met -> (D |` (R X. R)) e. Met)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960   i^i cin 2049   (_ wss 2050   X. cxp 3174  dom cdm 3176   |` cres 3178  -->wf 3184  RRcr 5245  Metcme 7786

This theorem is referenced by:  cncfmet 7902  remet 7907  lmsslem 7949  lmss 7950  caussi 7951  causs 7952  cmsss 7994

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-met 7790

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