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Theorem xkotop 23596
Description: The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
xkotop ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ Top)

Proof of Theorem xkotop
Dummy variables 𝑓 𝑘 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 𝑅 = 𝑅
2 eqid 2737 . . 3 {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}
3 eqid 2737 . . 3 (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
41, 2, 3xkoval 23595 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
5 fibas 22984 . . 3 (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ TopBases
6 tgcl 22976 . . 3 ((fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ TopBases → (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))) ∈ Top)
75, 6ax-mp 5 . 2 (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))) ∈ Top
84, 7eqeltrdi 2849 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  {crab 3436  wss 3951  𝒫 cpw 4600   cuni 4907  ran crn 5686  cima 5688  cfv 6561  (class class class)co 7431  cmpo 7433  ficfi 9450  t crest 17465  topGenctg 17482  Topctop 22899  TopBasesctb 22952   Cn ccn 23232  Compccmp 23394  ko cxko 23569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-en 8986  df-fin 8989  df-fi 9451  df-topgen 17488  df-top 22900  df-bases 22953  df-xko 23571
This theorem is referenced by:  xkotopon  23608  xkohaus  23661  xkoptsub  23662  xkococnlem  23667  xkococn  23668  xkohmeo  23823
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