Theorem isms 12625

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 5421	. . . . 5 |- ( f = K -> ( dist ` f ) = ( dist ` K ) )
2		fveq2 5421	. . . . . . 7 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
3		isms.x	. . . . . . 7 |- X = ( Base ` K )
4	2, 3	syl6eqr 2190	. . . . . 6 |- ( f = K -> ( Base ` f ) = X )
5	4	sqxpeqd 4565	. . . . 5 |- ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
6	1, 5	reseq12d 4820	. . . 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	syl6eqr 2190	. . 3 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
9	4	fveq2d 5425	. . 3 |- ( f = K -> ( Met ` ( Base ` f ) ) = ( Met ` X ) )
10	8, 9	eleq12d 2210	. 2 |- ( f = K -> ( ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) <-> D e. ( Met ` X ) ) )
11		df-ms 12512	. 2 |- MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
12	10, 11	elrab2 2843	1 |- ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   X. cxp 4537   |` cres 4541   ` cfv 5123   Basecbs 11962   distcds 12033   TopOpenctopn 12124   Metcmet 12153   *MetSpcxms 12508   MetSpcms 12509

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-rab 2425  df-v 2688  df-un 3075  df-in 3077  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-res 4551  df-iota 5088  df-fv 5131  df-ms 12512

This theorem is referenced by:  isms2  12626  msxms  12630  mspropd  12650