Theorem basmetres 10185

Description: The base set of a submetric. (Contributed by Jeff Hankins, 30-Aug-2009.)

Hypotheses

Ref	Expression
basmetres.1	|- D = (C |` (Y X. Y))
basmetres.2	|- X = dom dom C

Assertion

Ref	Expression
basmetres	|- ((C e. Met /\ Y C_ X) -> Y = dom dom D)

Proof of Theorem basmetres

Step	Hyp	Ref	Expression
1		xpss12 4089	. . . . . . . 8 |- ((Y C_ X /\ Y C_ X) -> (Y X. Y) C_ (X X. X))
2	1	anidms 480	. . . . . . 7 |- (Y C_ X -> (Y X. Y) C_ (X X. X))
3		df-ss 2605	. . . . . . 7 |- ((Y X. Y) C_ (X X. X) <-> ((Y X. Y) i^i (X X. X)) = (Y X. Y))
4	2, 3	sylib 215	. . . . . 6 |- (Y C_ X -> ((Y X. Y) i^i (X X. X)) = (Y X. Y))
5	4	dmeqd 4159	. . . . 5 |- (Y C_ X -> dom ((Y X. Y) i^i (X X. X)) = dom ( Y X. Y))
6	5	adantl 424	. . . 4 |- ((C e. Met /\ Y C_ X) -> dom ((Y X. Y) i^i (X X. X)) = dom ( Y X. Y))
7		dmxpid 4179	. . . 4 |- dom ( Y X. Y) = Y
8	6, 7	syl6req 1945	. . 3 |- ((C e. Met /\ Y C_ X) -> Y = dom ((Y X. Y) i^i (X X. X)))
9		basmetres.2	. . . . . . . . 9 |- X = dom dom C
10	9	metf 9084	. . . . . . . 8 |- (C e. Met -> C:(X X. X)-->RR)
11		fdm 4567	. . . . . . . 8 |- (C:(X X. X)-->RR -> dom C = (X X. X))
12	10, 11	syl 12	. . . . . . 7 |- (C e. Met -> dom C = (X X. X))
13	12	ineq2d 2796	. . . . . 6 |- (C e. Met -> ((Y X. Y) i^i dom C) = ((Y X. Y) i^i (X X. X)))
14	13	dmeqd 4159	. . . . 5 |- (C e. Met -> dom ((Y X. Y) i^i dom C) = dom ((Y X. Y) i^i (X X. X)))
15	14	eqcomd 1889	. . . 4 |- (C e. Met -> dom ((Y X. Y) i^i (X X. X)) = dom ((Y X. Y) i^i dom C))
16	15	adantr 425	. . 3 |- ((C e. Met /\ Y C_ X) -> dom ((Y X. Y) i^i (X X. X)) = dom ((Y X. Y) i^i dom C))
17	8, 16	eqtrd 1925	. 2 |- ((C e. Met /\ Y C_ X) -> Y = dom ((Y X. Y) i^i dom C))
18		basmetres.1	. . . . 5 |- D = (C |` (Y X. Y))
19	18	dmeqi 4158	. . . 4 |- dom D = dom ( C |` (Y X. Y))
20		dmres 4234	. . . 4 |- dom ( C |` (Y X. Y)) = ((Y X. Y) i^i dom C)
21	19, 20	eqtri 1908	. . 3 |- dom D = ((Y X. Y) i^i dom C)
22	21	dmeqi 4158	. 2 |- dom dom D = dom ((Y X. Y) i^i dom C)
23	17, 22	syl6eqr 1946	1 |- ((C e. Met /\ Y C_ X) -> Y = dom dom D)

Syntax hints:   -> wi 3   /\ wa 240   = 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:  subtopmetlem 10255  subtopmet 10256  blssp 15844  ismtyres 15954

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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