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Theorem xkoval 23595
Description: Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
xkoval.x 𝑋 = 𝑅
xkoval.k 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}
xkoval.t 𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
Assertion
Ref Expression
xkoval ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
Distinct variable groups:   𝑣,𝑘,𝐾   𝑓,𝑘,𝑣,𝑥,𝑅   𝑆,𝑓,𝑘,𝑣,𝑥   𝑘,𝑋,𝑥
Allowed substitution hints:   𝑇(𝑥,𝑣,𝑓,𝑘)   𝐾(𝑥,𝑓)   𝑋(𝑣,𝑓)

Proof of Theorem xkoval
Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . . . . . . . . . . 13 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
21unieqd 4920 . . . . . . . . . . . 12 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
3 xkoval.x . . . . . . . . . . . 12 𝑋 = 𝑅
42, 3eqtr4di 2795 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑋)
54pweqd 4617 . . . . . . . . . 10 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝒫 𝑟 = 𝒫 𝑋)
61oveq1d 7446 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑟t 𝑥) = (𝑅t 𝑥))
76eleq1d 2826 . . . . . . . . . 10 ((𝑠 = 𝑆𝑟 = 𝑅) → ((𝑟t 𝑥) ∈ Comp ↔ (𝑅t 𝑥) ∈ Comp))
85, 7rabeqbidv 3455 . . . . . . . . 9 ((𝑠 = 𝑆𝑟 = 𝑅) → {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp})
9 xkoval.k . . . . . . . . 9 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}
108, 9eqtr4di 2795 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp} = 𝐾)
11 simpl 482 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑠 = 𝑆)
121, 11oveq12d 7449 . . . . . . . . 9 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑟 Cn 𝑠) = (𝑅 Cn 𝑆))
1312rabeqdv 3452 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
1410, 11, 13mpoeq123dv 7508 . . . . . . 7 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
15 xkoval.t . . . . . . 7 𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
1614, 15eqtr4di 2795 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = 𝑇)
1716rneqd 5949 . . . . 5 ((𝑠 = 𝑆𝑟 = 𝑅) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = ran 𝑇)
1817fveq2d 6910 . . . 4 ((𝑠 = 𝑆𝑟 = 𝑅) → (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣})) = (fi‘ran 𝑇))
1918fveq2d 6910 . . 3 ((𝑠 = 𝑆𝑟 = 𝑅) → (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}))) = (topGen‘(fi‘ran 𝑇)))
20 df-xko 23571 . . 3 ko = (𝑠 ∈ Top, 𝑟 ∈ Top ↦ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}))))
21 fvex 6919 . . 3 (topGen‘(fi‘ran 𝑇)) ∈ V
2219, 20, 21ovmpoa 7588 . 2 ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
2322ancoms 458 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  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   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-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-xko 23571
This theorem is referenced by:  xkotop  23596  xkoopn  23597  xkouni  23607  xkoccn  23627  xkopt  23663  xkoco1cn  23665  xkoco2cn  23666  xkococn  23668  xkoinjcn  23695
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