Theorem tsettps 23252

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 23251	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))
4		id 23	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ (TopOn‘𝐴))
5	3, 4	eqeltrrd 2862	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
6		eqid 2761	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
7	1, 6	istps 23245	. 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
8	5, 7	sylibr 237	1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6537  Basecbs 17380  TopSetcts 17427  TopOpenctopn 17585  TopOnctopon 23221  TopSpctps 23243

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-rest 17586  df-topn 17587  df-top 23205  df-topon 23222  df-topsp 23244

This theorem is used by:  eltpsg  23254  indistpsALT  23324  xrstps  23520  prdstps  23941