Theorem isxms 15475

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 5690	. . . 4 |- ( f = K -> ( TopOpen ` f ) = ( TopOpen ` K ) )
2		isms.j	. . . 4 |- J = ( TopOpen ` K )
3	1, 2	eqtr4di 2289	. . 3 |- ( f = K -> ( TopOpen ` f ) = J )
4		fveq2 5690	. . . . . 6 |- ( f = K -> ( dist ` f ) = ( dist ` K ) )
5		fveq2 5690	. . . . . . . 8 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
6		isms.x	. . . . . . . 8 |- X = ( Base ` K )
7	5, 6	eqtr4di 2289	. . . . . . 7 |- ( f = K -> ( Base ` f ) = X )
8	7	sqxpeqd 4795	. . . . . 6 |- ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
9	4, 8	reseq12d 5059	. . . . 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 2289	. . . 4 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
12	11	fveq2d 5694	. . 3 |- ( f = K -> ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) = ( MetOpen ` D ) )
13	3, 12	eqeq12d 2253	. 2 |- ( f = K -> ( ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) <-> J = ( MetOpen ` D ) ) )
14		df-xms 15363	. 2 |- *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
15	13, 14	elrab2 2985	1 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   X. cxp 4767   |` cres 4771   ` cfv 5372   Basecbs 13330   distcds 13417   TopOpenctopn 13571   MetOpencmopn 14850   TopSpctps 15054   *MetSpcxms 15360

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-res 4781  df-iota 5332  df-fv 5380  df-xms 15363

This theorem is referenced by:  isxms2  15476  xmstopn  15479  xmstps  15481  xmspropd  15501