Theorem isxms 14923

Description: Express the predicate " <. X , D >. is an extended metric space" with underlying set X and distance function D. (Contributed by Mario Carneiro, 2-Sep-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
isxms	|- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )

Proof of Theorem isxms

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5576	. . . 4 |- ( f = K -> ( TopOpen ` f ) = ( TopOpen ` K ) )
2		isms.j	. . . 4 |- J = ( TopOpen ` K )
3	1, 2	eqtr4di 2256	. . 3 |- ( f = K -> ( TopOpen ` f ) = J )
4		fveq2 5576	. . . . . 6 |- ( f = K -> ( dist ` f ) = ( dist ` K ) )
5		fveq2 5576	. . . . . . . 8 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
6		isms.x	. . . . . . . 8 |- X = ( Base ` K )
7	5, 6	eqtr4di 2256	. . . . . . 7 |- ( f = K -> ( Base ` f ) = X )
8	7	sqxpeqd 4701	. . . . . 6 |- ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
9	4, 8	reseq12d 4960	. . . . 5 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = ( ( dist ` K ) |` ( X X. X ) ) )
10		isms.d	. . . . 5 |- D = ( ( dist ` K ) |` ( X X. X ) )
11	9, 10	eqtr4di 2256	. . . 4 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
12	11	fveq2d 5580	. . 3 |- ( f = K -> ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) = ( MetOpen ` D ) )
13	3, 12	eqeq12d 2220	. 2 |- ( f = K -> ( ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) <-> J = ( MetOpen ` D ) ) )
14		df-xms 14811	. 2 |- *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
15	13, 14	elrab2 2932	1 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   X. cxp 4673   |` cres 4677   ` cfv 5271   Basecbs 12832   distcds 12918   TopOpenctopn 13072   MetOpencmopn 14303   TopSpctps 14502   *MetSpcxms 14808

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-xms 14811

This theorem is referenced by:  isxms2  14924  xmstopn  14927  xmstps  14929  xmspropd  14949