Theorem tsettps 21237

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 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)

Proof of Theorem tsettps

Step	Hyp	Ref	Expression
1		tsettps.a	. . . 4 ⊢ 𝐴 = (Base‘𝐾)
2		tsettps.j	. . . 4 ⊢ 𝐽 = (TopSet‘𝐾)
3	1, 2	topontopn 21236	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))
4		id 22	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ (TopOn‘𝐴))
5	3, 4	eqeltrrd 2886	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
6		eqid 2797	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
7	1, 6	istps 21230	. 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
8	5, 7	sylibr 235	1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)

Syntax hints:   → wi 4   = wceq 1525   ∈ wcel 2083  ‘cfv 6232  Basecbs 16316  TopSetcts 16404  TopOpenctopn 16528  TopOnctopon 21206  TopSpctps 21228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-rest 16529  df-topn 16530  df-top 21190  df-topon 21207  df-topsp 21229

This theorem is referenced by:  eltpsg  21239  indistpsALT  21309  xrstps  21505  prdstps  21925