Theorem isxms 22252

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 6191	. . . 4 |- ( f = K -> ( TopOpen ` f ) = ( TopOpen ` K ) )
2		isms.j	. . . 4 |- J = ( TopOpen ` K )
3	1, 2	syl6eqr 2674	. . 3 |- ( f = K -> ( TopOpen ` f ) = J )
4		fveq2 6191	. . . . . 6 |- ( f = K -> ( dist ` f ) = ( dist ` K ) )
5		fveq2 6191	. . . . . . . 8 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
6		isms.x	. . . . . . . 8 |- X = ( Base ` K )
7	5, 6	syl6eqr 2674	. . . . . . 7 |- ( f = K -> ( Base ` f ) = X )
8	7	sqxpeqd 5141	. . . . . 6 |- ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
9	4, 8	reseq12d 5397	. . . . 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	syl6eqr 2674	. . . 4 |- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
12	11	fveq2d 6195	. . 3 |- ( f = K -> ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) = ( MetOpen ` D ) )
13	3, 12	eqeq12d 2637	. 2 |- ( f = K -> ( ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) <-> J = ( MetOpen ` D ) ) )
14		df-xms 22125	. 2 |- *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
15	13, 14	elrab2 3366	1 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   X. cxp 5112   |` cres 5116   ` cfv 5888   Basecbs 15857   distcds 15950   TopOpenctopn 16082   MetOpencmopn 19736   TopSpctps 20736   *MetSpcxme 22122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896  df-xms 22125

This theorem is referenced by:  isxms2  22253  xmstopn  22256  xmstps  22258  xmspropd  22278