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Theorem xkotop 22191
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 2822 . . 3 𝑅 = 𝑅
2 eqid 2822 . . 3 {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}
3 eqid 2822 . . 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 22190 . 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 21580 . . 3 (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ TopBases
6 tgcl 21572 . . 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 2922 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2114  {crab 3134  wss 3908  𝒫 cpw 4511   cuni 4813  ran crn 5533  cima 5535  cfv 6334  (class class class)co 7140  cmpo 7142  ficfi 8862  t crest 16685  topGenctg 16702  Topctop 21496  TopBasesctb 21548   Cn ccn 21827  Compccmp 21989  ko cxko 22164
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-oadd 8093  df-er 8276  df-en 8497  df-fin 8500  df-fi 8863  df-topgen 16708  df-top 21497  df-bases 21549  df-xko 22166
This theorem is referenced by:  xkotopon  22203  xkohaus  22256  xkoptsub  22257  xkococnlem  22262  xkococn  22263  xkohmeo  22418
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