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Theorem tsettps 17008
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 17007 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  J  =  ( TopOpen `  K )
)
4 id 20 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  J  e.  (TopOn `  A ) )
53, 4eqeltrrd 2511 . 2  |-  ( J  e.  (TopOn `  A
)  ->  ( TopOpen `  K )  e.  (TopOn `  A ) )
6 eqid 2436 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
71, 6istps 17001 . 2  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  A )
)
85, 7sylibr 204 1  |-  ( J  e.  (TopOn `  A
)  ->  K  e.  TopSp
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454   Basecbs 13469  TopSetcts 13535   TopOpenctopn 13649  TopOnctopon 16959   TopSpctps 16961
This theorem is referenced by:  eltpsg  17010  indistpsALT  17077  xrstps  17273  prdstps  17661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-rest 13650  df-topn 13651  df-top 16963  df-topon 16966  df-topsp 16967
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