Theorem resubmet 12467

Description: The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)

Hypotheses

Ref	Expression
resubmet.1	|- R = ( topGen ` ran (,) )
resubmet.2	|- J = ( MetOpen ` ( ( abs o. - ) |` ( A X. A ) ) )

Assertion

Ref	Expression
resubmet	|- ( A C_ RR -> J = ( R ↾t A ) )

Proof of Theorem resubmet

Step	Hyp	Ref	Expression
1		xpss12 4584	. . . . . 6 |- ( ( A C_ RR /\ A C_ RR ) -> ( A X. A ) C_ ( RR X. RR ) )
2	1	anidms 392	. . . . 5 |- ( A C_ RR -> ( A X. A ) C_ ( RR X. RR ) )
3	2	resabs1d 4785	. . . 4 |- ( A C_ RR -> ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( A X. A ) ) = ( ( abs o. - ) |` ( A X. A ) ) )
4	3	fveq2d 5357	. . 3 |- ( A C_ RR -> ( MetOpen ` ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( A X. A ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( A X. A ) ) ) )
5		resubmet.2	. . 3 |- J = ( MetOpen ` ( ( abs o. - ) |` ( A X. A ) ) )
6	4, 5	syl6reqr 2151	. 2 |- ( A C_ RR -> J = ( MetOpen ` ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( A X. A ) ) ) )
7		eqid 2100	. . . 4 |- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) )
8	7	rexmet 12460	. . 3 |- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR )
9		eqid 2100	. . . 4 |- ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( A X. A ) ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( A X. A ) )
10		resubmet.1	. . . . 5 |- R = ( topGen ` ran (,) )
11		eqid 2100	. . . . . 6 |- ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
12	7, 11	tgioo 12465	. . . . 5 |- ( topGen ` ran (,) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
13	10, 12	eqtri 2120	. . . 4 |- R = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
14		eqid 2100	. . . 4 |- ( MetOpen ` ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( A X. A ) ) ) = ( MetOpen ` ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( A X. A ) ) )
15	9, 13, 14	metrest 12434	. . 3 |- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ A C_ RR ) -> ( R ↾t A ) = ( MetOpen ` ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( A X. A ) ) ) )
16	8, 15	mpan 418	. 2 |- ( A C_ RR -> ( R ↾t A ) = ( MetOpen ` ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( A X. A ) ) ) )
17	6, 16	eqtr4d 2135	1 |- ( A C_ RR -> J = ( R ↾t A ) )

Syntax hints:   -> wi 4   = wceq 1299   e. wcel 1448   C_ wss 3021   X. cxp 4475   ran crn 4478   |` cres 4479   o. ccom 4481   ` cfv 5059  (class class class)co 5706   RRcr 7499   - cmin 7804   (,)cioo 9512   abscabs 10609   ↾_t crest 11902   topGenctg 11917   *Metcxmet 11931   MetOpencmopn 11936

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-map 6474  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-xneg 9400  df-xadd 9401  df-ioo 9516  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-rest 11904  df-topgen 11923  df-psmet 11938  df-xmet 11939  df-met 11940  df-bl 11941  df-mopn 11942  df-top 11947  df-topon 11960  df-bases 11992

This theorem is referenced by: (None)