Theorem setsmstsetg 14801

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 12890	. . . 4 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
4	3	setsslid 12754	. . 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 5565	. 2 |- ( ph -> (TopSet ` K ) = (TopSet ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
8	5, 7	eqtr4d 2232	1 |- ( ph -> ( MetOpen ` D ) = (TopSet ` K ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   <.cop 3626   X. cxp 4662   |` cres 4666   ` cfv 5259  (class class class)co 5925   ndxcnx 12700   sSet csts 12701   Basecbs 12703  TopSetcts 12786   distcds 12789   MetOpencmopn 14173

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-ndx 12706  df-slot 12707  df-sets 12710  df-tset 12799

This theorem is referenced by: (None)