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Theorem tsettps 16737
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 16736 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  J  =  ( TopOpen `  K )
)
4 id 19 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  J  e.  (TopOn `  A ) )
53, 4eqeltrrd 2391 . 2  |-  ( J  e.  (TopOn `  A
)  ->  ( TopOpen `  K )  e.  (TopOn `  A ) )
6 eqid 2316 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
71, 6istps 16730 . 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 1633    e. wcel 1701   ` cfv 5292   Basecbs 13195  TopSetcts 13261   TopOpenctopn 13375  TopOnctopon 16688   TopSpctps 16690
This theorem is referenced by:  eltpsg  16739  indistpsALT  16806  xrstps  16995  prdstps  17379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-rest 13376  df-topn 13377  df-top 16692  df-topon 16695  df-topsp 16696
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