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Theorem tsettps 12436
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 12435 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  J  =  ( TopOpen `  K )
)
4 id 19 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  J  e.  (TopOn `  A ) )
53, 4eqeltrrd 2235 . 2  |-  ( J  e.  (TopOn `  A
)  ->  ( TopOpen `  K )  e.  (TopOn `  A ) )
6 eqid 2157 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
71, 6istps 12430 . 2  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  A )
)
85, 7sylibr 133 1  |-  ( J  e.  (TopOn `  A
)  ->  K  e.  TopSp
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    e. wcel 2128   ` cfv 5170   Basecbs 12190  TopSetcts 12258   TopOpenctopn 12352  TopOnctopon 12408   TopSpctps 12428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1re 7826  ax-addrcl 7829
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-5 8895  df-6 8896  df-7 8897  df-8 8898  df-9 8899  df-ndx 12193  df-slot 12194  df-base 12196  df-tset 12271  df-rest 12353  df-topn 12354  df-top 12396  df-topon 12409  df-topsp 12429
This theorem is referenced by:  eltpsg  12438
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