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Theorem xkoval 21563
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 479 . . . . . . . . . . . . 13 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
21unieqd 4586 . . . . . . . . . . . 12 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
3 xkoval.x . . . . . . . . . . . 12 𝑋 = 𝑅
42, 3syl6eqr 2800 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑋)
54pweqd 4295 . . . . . . . . . 10 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝒫 𝑟 = 𝒫 𝑋)
61oveq1d 6816 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑟t 𝑥) = (𝑅t 𝑥))
76eleq1d 2812 . . . . . . . . . 10 ((𝑠 = 𝑆𝑟 = 𝑅) → ((𝑟t 𝑥) ∈ Comp ↔ (𝑅t 𝑥) ∈ Comp))
85, 7rabeqbidv 3323 . . . . . . . . 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 2800 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp} = 𝐾)
11 simpl 474 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑠 = 𝑆)
121, 11oveq12d 6819 . . . . . . . . 9 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑟 Cn 𝑠) = (𝑅 Cn 𝑆))
13 rabeq 3320 . . . . . . . . 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 6870 . . . . . . 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 2800 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = 𝑇)
1817rneqd 5496 . . . . 5 ((𝑠 = 𝑆𝑟 = 𝑅) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}) = ran 𝑇)
1918fveq2d 6344 . . . 4 ((𝑠 = 𝑆𝑟 = 𝑅) → (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣})) = (fi‘ran 𝑇))
2019fveq2d 6344 . . 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 21539 . . 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 6350 . . 3 (topGen‘(fi‘ran 𝑇)) ∈ V
2320, 21, 22ovmpt2a 6944 . 2 ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
2423ancoms 468 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wcel 2127  {crab 3042  wss 3703  𝒫 cpw 4290   cuni 4576  ran crn 5255  cima 5257  cfv 6037  (class class class)co 6801  cmpt2 6803  ficfi 8469  t crest 16254  topGenctg 16271  Topctop 20871   Cn ccn 21201  Compccmp 21362   ^ko cxko 21537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-xko 21539
This theorem is referenced by:  xkotop  21564  xkoopn  21565  xkouni  21575  xkoccn  21595  xkopt  21631  xkoco1cn  21633  xkoco2cn  21634  xkococn  21636  xkoinjcn  21663
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