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Theorem xkoval 22123
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 485 . . . . . . . . . . . . 13 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
21unieqd 4840 . . . . . . . . . . . 12 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
3 xkoval.x . . . . . . . . . . . 12 𝑋 = 𝑅
42, 3syl6eqr 2871 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑋)
54pweqd 4540 . . . . . . . . . 10 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝒫 𝑟 = 𝒫 𝑋)
61oveq1d 7160 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑟t 𝑥) = (𝑅t 𝑥))
76eleq1d 2894 . . . . . . . . . 10 ((𝑠 = 𝑆𝑟 = 𝑅) → ((𝑟t 𝑥) ∈ Comp ↔ (𝑅t 𝑥) ∈ Comp))
85, 7rabeqbidv 3483 . . . . . . . . 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, 9syl6eqr 2871 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp} = 𝐾)
11 simpl 483 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑠 = 𝑆)
121, 11oveq12d 7163 . . . . . . . . 9 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑟 Cn 𝑠) = (𝑅 Cn 𝑆))
1312rabeqdv 3482 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
1410, 11, 13mpoeq123dv 7218 . . . . . . 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, 15syl6eqr 2871 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = 𝑇)
1716rneqd 5801 . . . . 5 ((𝑠 = 𝑆𝑟 = 𝑅) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = ran 𝑇)
1817fveq2d 6667 . . . 4 ((𝑠 = 𝑆𝑟 = 𝑅) → (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣})) = (fi‘ran 𝑇))
1918fveq2d 6667 . . 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 22099 . . 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 6676 . . 3 (topGen‘(fi‘ran 𝑇)) ∈ V
2219, 20, 21ovmpoa 7294 . 2 ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
2322ancoms 459 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  wss 3933  𝒫 cpw 4535   cuni 4830  ran crn 5549  cima 5551  cfv 6348  (class class class)co 7145  cmpo 7147  ficfi 8862  t crest 16682  topGenctg 16699  Topctop 21429   Cn ccn 21760  Compccmp 21922  ko cxko 22097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-xko 22099
This theorem is referenced by:  xkotop  22124  xkoopn  22125  xkouni  22135  xkoccn  22155  xkopt  22191  xkoco1cn  22193  xkoco2cn  22194  xkococn  22196  xkoinjcn  22223
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