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Theorem setsmstsetg 13843
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 12634 . . . 4  |-  (TopSet  = Slot  (TopSet `  ndx )  /\  (TopSet `  ndx )  e.  NN )
43setsslid 12504 . . 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 5517 . 2  |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  ( M sSet  <.
(TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) ) )
85, 7eqtr4d 2213 1  |-  ( ph  ->  ( MetOpen `  D )  =  (TopSet `  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   <.cop 3595    X. cxp 4623    |` cres 4627   ` cfv 5214  (class class class)co 5871   ndxcnx 12450   sSet csts 12451   Basecbs 12453  TopSetcts 12533   distcds 12536   MetOpencmopn 13305
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 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898  ax-resscn 7899  ax-1re 7901  ax-addrcl 7904
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5176  df-fun 5216  df-fv 5222  df-ov 5874  df-oprab 5875  df-mpo 5876  df-inn 8915  df-2 8973  df-3 8974  df-4 8975  df-5 8976  df-6 8977  df-7 8978  df-8 8979  df-9 8980  df-ndx 12456  df-slot 12457  df-sets 12460  df-tset 12546
This theorem is referenced by: (None)
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