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Theorem xkoval 21300
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 477 . . . . . . . . . . . . 13 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
21unieqd 4412 . . . . . . . . . . . 12 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
3 xkoval.x . . . . . . . . . . . 12 𝑋 = 𝑅
42, 3syl6eqr 2673 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑋)
54pweqd 4135 . . . . . . . . . 10 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝒫 𝑟 = 𝒫 𝑋)
61oveq1d 6619 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑟t 𝑥) = (𝑅t 𝑥))
76eleq1d 2683 . . . . . . . . . 10 ((𝑠 = 𝑆𝑟 = 𝑅) → ((𝑟t 𝑥) ∈ Comp ↔ (𝑅t 𝑥) ∈ Comp))
85, 7rabeqbidv 3181 . . . . . . . . 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 2673 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp} = 𝐾)
11 simpl 473 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑠 = 𝑆)
121, 11oveq12d 6622 . . . . . . . . 9 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑟 Cn 𝑠) = (𝑅 Cn 𝑆))
13 rabeq 3179 . . . . . . . . 9 ((𝑟 Cn 𝑠) = (𝑅 Cn 𝑆) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
1412, 13syl 17 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
1510, 11, 14mpt2eq123dv 6670 . . . . . . 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}))
16 xkoval.t . . . . . . 7 𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
1715, 16syl6eqr 2673 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = 𝑇)
1817rneqd 5313 . . . . 5 ((𝑠 = 𝑆𝑟 = 𝑅) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = ran 𝑇)
1918fveq2d 6152 . . . 4 ((𝑠 = 𝑆𝑟 = 𝑅) → (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣})) = (fi‘ran 𝑇))
2019fveq2d 6152 . . 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)))
21 df-xko 21276 . . 3 ^ko = (𝑠 ∈ Top, 𝑟 ∈ Top ↦ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}))))
22 fvex 6158 . . 3 (topGen‘(fi‘ran 𝑇)) ∈ V
2320, 21, 22ovmpt2a 6744 . 2 ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
2423ancoms 469 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  wss 3555  𝒫 cpw 4130   cuni 4402  ran crn 5075  cima 5077  cfv 5847  (class class class)co 6604  cmpt2 6606  ficfi 8260  t crest 16002  topGenctg 16019  Topctop 20617   Cn ccn 20938  Compccmp 21099   ^ko cxko 21274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-xko 21276
This theorem is referenced by:  xkotop  21301  xkoopn  21302  xkouni  21312  xkoccn  21332  xkopt  21368  xkoco1cn  21370  xkoco2cn  21371  xkococn  21373  xkoinjcn  21400
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