Theorem isxms 15125

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 5627	. . . 4 |- ( f = K -> ( TopOpen ` f ) = ( TopOpen ` K ) )
2		isms.j	. . . 4 |- J = ( TopOpen ` K )
3	1, 2	eqtr4di 2280	. . 3 |- ( f = K -> ( TopOpen ` f ) = J )
4		fveq2 5627	. . . . . 6 |- ( f = K -> ( dist ` f ) = ( dist ` K ) )
5		fveq2 5627	. . . . . . . 8 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
6		isms.x	. . . . . . . 8 |- X = ( Base ` K )
7	5, 6	eqtr4di 2280	. . . . . . 7 |- ( f = K -> ( Base ` f ) = X )
8	7	sqxpeqd 4745	. . . . . 6 |- ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
9	4, 8	reseq12d 5006	. . . . 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 2280	. . . 4 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
12	11	fveq2d 5631	. . 3 |- ( f = K -> ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) = ( MetOpen ` D ) )
13	3, 12	eqeq12d 2244	. 2 |- ( f = K -> ( ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) <-> J = ( MetOpen ` D ) ) )
14		df-xms 15013	. 2 |- *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
15	13, 14	elrab2 2962	1 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   X. cxp 4717   |` cres 4721   ` cfv 5318   Basecbs 13032   distcds 13119   TopOpenctopn 13273   MetOpencmopn 14505   TopSpctps 14704   *MetSpcxms 15010

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-res 4731  df-iota 5278  df-fv 5326  df-xms 15013

This theorem is referenced by:  isxms2  15126  xmstopn  15129  xmstps  15131  xmspropd  15151