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Theorem tsettps 18673
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 18672 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  J  =  ( TopOpen `  K )
)
4 id 22 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  J  e.  (TopOn `  A ) )
53, 4eqeltrrd 2540 . 2  |-  ( J  e.  (TopOn `  A
)  ->  ( TopOpen `  K )  e.  (TopOn `  A ) )
6 eqid 2451 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
71, 6istps 18666 . 2  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  A )
)
85, 7sylibr 212 1  |-  ( J  e.  (TopOn `  A
)  ->  K  e.  TopSp
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   ` cfv 5519   Basecbs 14285  TopSetcts 14355   TopOpenctopn 14471  TopOnctopon 18624   TopSpctps 18626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-rest 14472  df-topn 14473  df-top 18628  df-topon 18631  df-topsp 18632
This theorem is referenced by:  eltpsg  18675  indistpsALT  18742  xrstps  18938  prdstps  19327
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