Theorem setsmstsetg 15176

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 13242	. . . 4 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
4	3	setsslid 13104	. . 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 5636	. 2 |- ( ph -> (TopSet ` K ) = (TopSet ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
8	5, 7	eqtr4d 2265	1 |- ( ph -> ( MetOpen ` D ) = (TopSet ` K ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   <.cop 3669   X. cxp 4718   |` cres 4722   ` cfv 5321  (class class class)co 6010   ndxcnx 13050   sSet csts 13051   Basecbs 13053  TopSetcts 13137   distcds 13140   MetOpencmopn 14526

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1re 8109  ax-addrcl 8112

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-iota 5281  df-fun 5323  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-ndx 13056  df-slot 13057  df-sets 13060  df-tset 13150

This theorem is referenced by: (None)