Theorem istps2 22436

Description: Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)

Hypotheses

Ref	Expression
istps.a	⊢ 𝐴 = (Base‘𝐾)
istps.j	⊢ 𝐽 = (TopOpen‘𝐾)

Assertion

Ref	Expression
istps2	⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽))

Proof of Theorem istps2

Step	Hyp	Ref	Expression
1		istps.a	. . 3 ⊢ 𝐴 = (Base‘𝐾)
2		istps.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐾)
3	1, 2	istps 22435	. 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))
4		istopon 22413	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽))
5	3, 4	bitri 274	1 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∪ cuni 4908  ‘cfv 6543  Basecbs 17143  TopOpenctopn 17366  Topctop 22394  TopOnctopon 22411  TopSpctps 22433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-top 22395  df-topon 22412  df-topsp 22434

This theorem is referenced by:  tpsuni  22437  tpstop  22438  istpsi  22443