Theorem setsmstsetg 14066

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 12648	. . . 4 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
4	3	setsslid 12515	. . 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 5521	. 2 |- ( ph -> (TopSet ` K ) = (TopSet ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
8	5, 7	eqtr4d 2213	1 |- ( ph -> ( MetOpen ` D ) = (TopSet ` K ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   <.cop 3597   X. cxp 4626   |` cres 4630   ` cfv 5218  (class class class)co 5877   ndxcnx 12461   sSet csts 12462   Basecbs 12464  TopSetcts 12544   distcds 12547   MetOpencmopn 13530

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-7 8985  df-8 8986  df-9 8987  df-ndx 12467  df-slot 12468  df-sets 12471  df-tset 12557

This theorem is referenced by: (None)