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Theorem setsmstsetg 14649
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
StepHypRef Expression
1 setsmsbasg.m . . 3  |-  ( ph  ->  M  e.  V )
2 setsmsbasg.d . . 3  |-  ( ph  ->  ( MetOpen `  D )  e.  W )
3 tsetslid 12805 . . . 4  |-  (TopSet  = Slot  (TopSet `  ndx )  /\  (TopSet `  ndx )  e.  NN )
43setsslid 12669 . . 3  |-  ( ( M  e.  V  /\  ( MetOpen `  D )  e.  W )  ->  ( MetOpen
`  D )  =  (TopSet `  ( M sSet  <.
(TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) ) )
51, 2, 4syl2anc 411 . 2  |-  ( ph  ->  ( MetOpen `  D )  =  (TopSet `  ( M sSet  <.
(TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) ) )
6 setsms.k . . 3  |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
76fveq2d 5558 . 2  |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  ( M sSet  <.
(TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) ) )
85, 7eqtr4d 2229 1  |-  ( ph  ->  ( MetOpen `  D )  =  (TopSet `  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2164   <.cop 3621    X. cxp 4657    |` cres 4661   ` cfv 5254  (class class class)co 5918   ndxcnx 12615   sSet csts 12616   Basecbs 12618  TopSetcts 12701   distcds 12704   MetOpencmopn 14037
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 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969
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 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-9 9048  df-ndx 12621  df-slot 12622  df-sets 12625  df-tset 12714
This theorem is referenced by: (None)
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