Theorem isms 14621

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 5554	. . . . 5 |- ( f = K -> ( dist ` f ) = ( dist ` K ) )
2		fveq2 5554	. . . . . . 7 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
3		isms.x	. . . . . . 7 |- X = ( Base ` K )
4	2, 3	eqtr4di 2244	. . . . . 6 |- ( f = K -> ( Base ` f ) = X )
5	4	sqxpeqd 4685	. . . . 5 |- ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
6	1, 5	reseq12d 4943	. . . 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 2244	. . 3 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
9	4	fveq2d 5558	. . 3 |- ( f = K -> ( Met ` ( Base ` f ) ) = ( Met ` X ) )
10	8, 9	eleq12d 2264	. 2 |- ( f = K -> ( ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) <-> D e. ( Met ` X ) ) )
11		df-ms 14508	. 2 |- MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
12	10, 11	elrab2 2919	1 |- ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   X. cxp 4657   |` cres 4661   ` cfv 5254   Basecbs 12618   distcds 12704   TopOpenctopn 12851   Metcmet 14033   *MetSpcxms 14504   MetSpcms 14505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-res 4671  df-iota 5215  df-fv 5262  df-ms 14508

This theorem is referenced by:  isms2  14622  msxms  14626  mspropd  14646