Theorem tsettps 22998

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 22997	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))
4		id 22	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ (TopOn‘𝐴))
5	3, 4	eqeltrrd 2863	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
6		eqid 2762	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
7	1, 6	istps 22991	. 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
8	5, 7	sylibr 236	1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  ‘cfv 6521  Basecbs 17245  TopSetcts 17292  TopOpenctopn 17450  TopOnctopon 22967  TopSpctps 22989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-rest 17451  df-topn 17452  df-top 22951  df-topon 22968  df-topsp 22990

This theorem is referenced by:  eltpsg  23000  indistpsALT  23070  xrstps  23266  prdstps  23686