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Theorem xkoval 21883
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) → (𝑆 ^ko 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 4761 . . . . . . . . . . . 12 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
3 xkoval.x . . . . . . . . . . . 12 𝑋 = 𝑅
42, 3syl6eqr 2851 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑋)
54pweqd 4464 . . . . . . . . . 10 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝒫 𝑟 = 𝒫 𝑋)
61oveq1d 7038 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑟t 𝑥) = (𝑅t 𝑥))
76eleq1d 2869 . . . . . . . . . 10 ((𝑠 = 𝑆𝑟 = 𝑅) → ((𝑟t 𝑥) ∈ Comp ↔ (𝑅t 𝑥) ∈ Comp))
85, 7rabeqbidv 3433 . . . . . . . . 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 2851 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp} = 𝐾)
11 simpl 483 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑠 = 𝑆)
121, 11oveq12d 7041 . . . . . . . . 9 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑟 Cn 𝑠) = (𝑅 Cn 𝑆))
1312rabeqdv 3432 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
1410, 11, 13mpoeq123dv 7094 . . . . . . 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 2851 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = 𝑇)
1716rneqd 5697 . . . . 5 ((𝑠 = 𝑆𝑟 = 𝑅) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = ran 𝑇)
1817fveq2d 6549 . . . 4 ((𝑠 = 𝑆𝑟 = 𝑅) → (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣})) = (fi‘ran 𝑇))
1918fveq2d 6549 . . 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 21859 . . 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 6558 . . 3 (topGen‘(fi‘ran 𝑇)) ∈ V
2219, 20, 21ovmpoa 7168 . 2 ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
2322ancoms 459 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wcel 2083  {crab 3111  wss 3865  𝒫 cpw 4459   cuni 4751  ran crn 5451  cima 5453  cfv 6232  (class class class)co 7023  cmpo 7025  ficfi 8727  t crest 16527  topGenctg 16544  Topctop 21189   Cn ccn 21520  Compccmp 21682   ^ko cxko 21857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-iota 6196  df-fun 6234  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-xko 21859
This theorem is referenced by:  xkotop  21884  xkoopn  21885  xkouni  21895  xkoccn  21915  xkopt  21951  xkoco1cn  21953  xkoco2cn  21954  xkococn  21956  xkoinjcn  21983
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