Theorem setsmstsetg 14660

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 12808	. . . 4 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
4	3	setsslid 12672	. . 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 5559	. 2 |- ( ph -> (TopSet ` K ) = (TopSet ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
8	5, 7	eqtr4d 2229	1 |- ( ph -> ( MetOpen ` D ) = (TopSet ` K ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   <.cop 3622   X. cxp 4658   |` cres 4662   ` cfv 5255  (class class class)co 5919   ndxcnx 12618   sSet csts 12619   Basecbs 12621  TopSetcts 12704   distcds 12707   MetOpencmopn 14040

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-ndx 12624  df-slot 12625  df-sets 12628  df-tset 12717

This theorem is referenced by: (None)