Theorem setsmstsetg 14606

Description: The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.)

Hypotheses

Ref	Expression
setsms.x	|- ( ph -> X = ( Base ` M ) )
setsms.d	|- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
setsms.k	|- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
setsmsbasg.m	|- ( ph -> M e. V )
setsmsbasg.d	|- ( ph -> ( MetOpen ` D ) e. W )

Assertion

Ref	Expression
setsmstsetg	|- ( ph -> ( MetOpen ` D ) = (TopSet ` K ) )

Proof of Theorem setsmstsetg

Step	Hyp	Ref	Expression
1		setsmsbasg.m	. . 3 |- ( ph -> M e. V )
2		setsmsbasg.d	. . 3 |- ( ph -> ( MetOpen ` D ) e. W )
3		tsetslid 12779	. . . 4 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
4	3	setsslid 12643	. . 3 |- ( ( M e. V /\ ( MetOpen ` D ) e. W ) -> ( MetOpen ` D ) = (TopSet ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
5	1, 2, 4	syl2anc 411	. 2 |- ( ph -> ( MetOpen ` D ) = (TopSet ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
6		setsms.k	. . 3 |- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
7	6	fveq2d 5546	. 2 |- ( ph -> (TopSet ` K ) = (TopSet ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
8	5, 7	eqtr4d 2225	1 |- ( ph -> ( MetOpen ` D ) = (TopSet ` K ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   <.cop 3617   X. cxp 4649   |` cres 4653   ` cfv 5242  (class class class)co 5906   ndxcnx 12589   sSet csts 12590   Basecbs 12592  TopSetcts 12675   distcds 12678   MetOpencmopn 14001

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-cnex 7949  ax-resscn 7950  ax-1re 7952  ax-addrcl 7955

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2758  df-sbc 2982  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-br 4026  df-opab 4087  df-mpt 4088  df-id 4318  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-iota 5203  df-fun 5244  df-fv 5250  df-ov 5909  df-oprab 5910  df-mpo 5911  df-inn 8969  df-2 9027  df-3 9028  df-4 9029  df-5 9030  df-6 9031  df-7 9032  df-8 9033  df-9 9034  df-ndx 12595  df-slot 12596  df-sets 12599  df-tset 12688

This theorem is referenced by: (None)