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Theorem setsmstsetg 12464
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 11946 . . . 4  |-  (TopSet  = Slot  (TopSet `  ndx )  /\  (TopSet `  ndx )  e.  NN )
43setsslid 11846 . . 3  |-  ( ( M  e.  V  /\  ( MetOpen `  D )  e.  W )  ->  ( MetOpen
`  D )  =  (TopSet `  ( M sSet  <.
(TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) ) )
51, 2, 4syl2anc 406 . 2  |-  ( ph  ->  ( MetOpen `  D )  =  (TopSet `  ( M sSet  <.
(TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) ) )
6 setsms.k . . 3  |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
76fveq2d 5377 . 2  |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  ( M sSet  <.
(TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) ) )
85, 7eqtr4d 2148 1  |-  ( ph  ->  ( MetOpen `  D )  =  (TopSet `  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1312    e. wcel 1461   <.cop 3494    X. cxp 4495    |` cres 4499   ` cfv 5079  (class class class)co 5726   ndxcnx 11793   sSet csts 11794   Basecbs 11796  TopSetcts 11864   distcds 11867   MetOpencmopn 11991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1re 7633  ax-addrcl 7636
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-iota 5044  df-fun 5081  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-inn 8625  df-2 8683  df-3 8684  df-4 8685  df-5 8686  df-6 8687  df-7 8688  df-8 8689  df-9 8690  df-ndx 11799  df-slot 11800  df-sets 11803  df-tset 11877
This theorem is referenced by: (None)
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