Theorem isms 13093

Description: Express the predicate " <. X , D >. is a metric space" with underlying set X and distance function D. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)

Hypotheses

Ref	Expression
isms.j	|- J = ( TopOpen ` K )
isms.x	|- X = ( Base ` K )
isms.d	|- D = ( ( dist ` K ) |` ( X X. X ) )

Assertion

Ref	Expression
isms	|- ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )

Proof of Theorem isms

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5486	. . . . 5 |- ( f = K -> ( dist ` f ) = ( dist ` K ) )
2		fveq2 5486	. . . . . . 7 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
3		isms.x	. . . . . . 7 |- X = ( Base ` K )
4	2, 3	eqtr4di 2217	. . . . . 6 |- ( f = K -> ( Base ` f ) = X )
5	4	sqxpeqd 4630	. . . . 5 |- ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
6	1, 5	reseq12d 4885	. . . 4 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = ( ( dist ` K ) |` ( X X. X ) ) )
7		isms.d	. . . 4 |- D = ( ( dist ` K ) |` ( X X. X ) )
8	6, 7	eqtr4di 2217	. . 3 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
9	4	fveq2d 5490	. . 3 |- ( f = K -> ( Met ` ( Base ` f ) ) = ( Met ` X ) )
10	8, 9	eleq12d 2237	. 2 |- ( f = K -> ( ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) <-> D e. ( Met ` X ) ) )
11		df-ms 12980	. 2 |- MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
12	10, 11	elrab2 2885	1 |- ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   X. cxp 4602   |` cres 4606   ` cfv 5188   Basecbs 12394   distcds 12466   TopOpenctopn 12557   Metcmet 12621   *MetSpcxms 12976   MetSpcms 12977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-rab 2453  df-v 2728  df-un 3120  df-in 3122  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-res 4616  df-iota 5153  df-fv 5196  df-ms 12980

This theorem is referenced by:  isms2  13094  msxms  13098  mspropd  13118