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Theorem tsettps 16681
Description: If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tsettps.a  |-  A  =  ( Base `  K
)
tsettps.j  |-  J  =  (TopSet `  K )
Assertion
Ref Expression
tsettps  |-  ( J  e.  (TopOn `  A
)  ->  K  e.  TopSp
)

Proof of Theorem tsettps
StepHypRef Expression
1 tsettps.a . . . 4  |-  A  =  ( Base `  K
)
2 tsettps.j . . . 4  |-  J  =  (TopSet `  K )
31, 2topontopn 16680 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  J  =  ( TopOpen `  K )
)
4 id 19 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  J  e.  (TopOn `  A ) )
53, 4eqeltrrd 2358 . 2  |-  ( J  e.  (TopOn `  A
)  ->  ( TopOpen `  K )  e.  (TopOn `  A ) )
6 eqid 2283 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
71, 6istps 16674 . 2  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  A )
)
85, 7sylibr 203 1  |-  ( J  e.  (TopOn `  A
)  ->  K  e.  TopSp
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255   Basecbs 13148  TopSetcts 13214   TopOpenctopn 13326  TopOnctopon 16632   TopSpctps 16634
This theorem is referenced by:  eltpsg  16683  indistpsALT  16750  xrstps  16939  prdstps  17323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-rest 13327  df-topn 13328  df-top 16636  df-topon 16639  df-topsp 16640
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