Theorem isxms 13245

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 5496	. . . 4 |- ( f = K -> ( TopOpen ` f ) = ( TopOpen ` K ) )
2		isms.j	. . . 4 |- J = ( TopOpen ` K )
3	1, 2	eqtr4di 2221	. . 3 |- ( f = K -> ( TopOpen ` f ) = J )
4		fveq2 5496	. . . . . 6 |- ( f = K -> ( dist ` f ) = ( dist ` K ) )
5		fveq2 5496	. . . . . . . 8 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
6		isms.x	. . . . . . . 8 |- X = ( Base ` K )
7	5, 6	eqtr4di 2221	. . . . . . 7 |- ( f = K -> ( Base ` f ) = X )
8	7	sqxpeqd 4637	. . . . . 6 |- ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
9	4, 8	reseq12d 4892	. . . . 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 2221	. . . 4 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
12	11	fveq2d 5500	. . 3 |- ( f = K -> ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) = ( MetOpen ` D ) )
13	3, 12	eqeq12d 2185	. 2 |- ( f = K -> ( ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) <-> J = ( MetOpen ` D ) ) )
14		df-xms 13133	. 2 |- *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
15	13, 14	elrab2 2889	1 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   X. cxp 4609   |` cres 4613   ` cfv 5198   Basecbs 12416   distcds 12489   TopOpenctopn 12580   MetOpencmopn 12779   TopSpctps 12822   *MetSpcxms 13130

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-res 4623  df-iota 5160  df-fv 5206  df-xms 13133

This theorem is referenced by:  isxms2  13246  xmstopn  13249  xmstps  13251  xmspropd  13271