Theorem isms 14925

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 5576	. . . . 5 |- ( f = K -> ( dist ` f ) = ( dist ` K ) )
2		fveq2 5576	. . . . . . 7 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
3		isms.x	. . . . . . 7 |- X = ( Base ` K )
4	2, 3	eqtr4di 2256	. . . . . 6 |- ( f = K -> ( Base ` f ) = X )
5	4	sqxpeqd 4701	. . . . 5 |- ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
6	1, 5	reseq12d 4960	. . . 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 2256	. . 3 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
9	4	fveq2d 5580	. . 3 |- ( f = K -> ( Met ` ( Base ` f ) ) = ( Met ` X ) )
10	8, 9	eleq12d 2276	. 2 |- ( f = K -> ( ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) <-> D e. ( Met ` X ) ) )
11		df-ms 14812	. 2 |- MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
12	10, 11	elrab2 2932	1 |- ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   X. cxp 4673   |` cres 4677   ` cfv 5271   Basecbs 12832   distcds 12918   TopOpenctopn 13072   Metcmet 14299   *MetSpcxms 14808   MetSpcms 14809

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-in 3172  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-res 4687  df-iota 5232  df-fv 5279  df-ms 14812

This theorem is referenced by:  isms2  14926  msxms  14930  mspropd  14950