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Theorem xkouni 23607
Description: The base set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
xkouni.1 𝐽 = (𝑆ko 𝑅)
Assertion
Ref Expression
xkouni ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = 𝐽)

Proof of Theorem xkouni
Dummy variables 𝑓 𝑘 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ima0 6095 . . . . . . . . 9 (𝑓 “ ∅) = ∅
2 0ss 4400 . . . . . . . . 9 ∅ ⊆ 𝑆
31, 2eqsstri 4030 . . . . . . . 8 (𝑓 “ ∅) ⊆ 𝑆
43a1i 11 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (𝑓 “ ∅) ⊆ 𝑆)
54ralrimiva 3146 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ 𝑆)
6 rabid2 3470 . . . . . 6 ((𝑅 Cn 𝑆) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ 𝑆} ↔ ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ 𝑆)
75, 6sylibr 234 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ 𝑆})
8 eqid 2737 . . . . . 6 𝑅 = 𝑅
9 simpl 482 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
10 simpr 484 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
11 0ss 4400 . . . . . . 7 ∅ ⊆ 𝑅
1211a1i 11 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∅ ⊆ 𝑅)
13 rest0 23177 . . . . . . . 8 (𝑅 ∈ Top → (𝑅t ∅) = {∅})
1413adantr 480 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅t ∅) = {∅})
15 0cmp 23402 . . . . . . 7 {∅} ∈ Comp
1614, 15eqeltrdi 2849 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅t ∅) ∈ Comp)
17 eqid 2737 . . . . . . . 8 𝑆 = 𝑆
1817topopn 22912 . . . . . . 7 (𝑆 ∈ Top → 𝑆𝑆)
1918adantl 481 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆𝑆)
208, 9, 10, 12, 16, 19xkoopn 23597 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ 𝑆} ∈ (𝑆ko 𝑅))
217, 20eqeltrd 2841 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆ko 𝑅))
22 xkouni.1 . . . 4 𝐽 = (𝑆ko 𝑅)
2321, 22eleqtrrdi 2852 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ 𝐽)
24 elssuni 4937 . . 3 ((𝑅 Cn 𝑆) ∈ 𝐽 → (𝑅 Cn 𝑆) ⊆ 𝐽)
2523, 24syl 17 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ⊆ 𝐽)
26 eqid 2737 . . . . . 6 {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}
27 eqid 2737 . . . . . 6 (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
288, 26, 27xkoval 23595 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
2928unieqd 4920 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
3022unieqi 4919 . . . 4 𝐽 = (𝑆ko 𝑅)
31 ovex 7464 . . . . . . . 8 (𝑅 Cn 𝑆) ∈ V
3231pwex 5380 . . . . . . 7 𝒫 (𝑅 Cn 𝑆) ∈ V
338, 26, 27xkotf 23593 . . . . . . . 8 (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}):({𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
34 frn 6743 . . . . . . . 8 ((𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}):({𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆))
3533, 34ax-mp 5 . . . . . . 7 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆)
3632, 35ssexi 5322 . . . . . 6 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V
37 fiuni 9468 . . . . . 6 (ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})))
3836, 37ax-mp 5 . . . . 5 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
39 fvex 6919 . . . . . 6 (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ V
40 unitg 22974 . . . . . 6 ((fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ V → (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))) = (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})))
4139, 40ax-mp 5 . . . . 5 (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))) = (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
4238, 41eqtr4i 2768 . . . 4 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})))
4329, 30, 423eqtr4g 2802 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 = ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
4435a1i 11 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆))
45 sspwuni 5100 . . . 4 (ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆) ↔ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ (𝑅 Cn 𝑆))
4644, 45sylib 218 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ (𝑅 Cn 𝑆))
4743, 46eqsstrd 4018 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ⊆ (𝑅 Cn 𝑆))
4825, 47eqssd 4001 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907   × cxp 5683  ran crn 5686  cima 5688  wf 6557  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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-xko 23571
This theorem is referenced by:  xkotopon  23608  xkohaus  23661  xkoptsub  23662
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