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Theorem xkouni 23623
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 6097 . . . . . . . . 9 (𝑓 “ ∅) = ∅
2 0ss 4406 . . . . . . . . 9 ∅ ⊆ 𝑆
31, 2eqsstri 4030 . . . . . . . 8 (𝑓 “ ∅) ⊆ 𝑆
43a1i 11 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (𝑓 “ ∅) ⊆ 𝑆)
54ralrimiva 3144 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ 𝑆)
6 rabid2 3468 . . . . . 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 2735 . . . . . 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 4406 . . . . . . 7 ∅ ⊆ 𝑅
1211a1i 11 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∅ ⊆ 𝑅)
13 rest0 23193 . . . . . . . 8 (𝑅 ∈ Top → (𝑅t ∅) = {∅})
1413adantr 480 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅t ∅) = {∅})
15 0cmp 23418 . . . . . . 7 {∅} ∈ Comp
1614, 15eqeltrdi 2847 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅t ∅) ∈ Comp)
17 eqid 2735 . . . . . . . 8 𝑆 = 𝑆
1817topopn 22928 . . . . . . 7 (𝑆 ∈ Top → 𝑆𝑆)
1918adantl 481 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆𝑆)
208, 9, 10, 12, 16, 19xkoopn 23613 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ 𝑆} ∈ (𝑆ko 𝑅))
217, 20eqeltrd 2839 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆ko 𝑅))
22 xkouni.1 . . . 4 𝐽 = (𝑆ko 𝑅)
2321, 22eleqtrrdi 2850 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ 𝐽)
24 elssuni 4942 . . 3 ((𝑅 Cn 𝑆) ∈ 𝐽 → (𝑅 Cn 𝑆) ⊆ 𝐽)
2523, 24syl 17 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ⊆ 𝐽)
26 eqid 2735 . . . . . 6 {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}
27 eqid 2735 . . . . . 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 23611 . . . . 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 4925 . . . 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 4924 . . . 4 𝐽 = (𝑆ko 𝑅)
31 ovex 7464 . . . . . . . 8 (𝑅 Cn 𝑆) ∈ V
3231pwex 5386 . . . . . . 7 𝒫 (𝑅 Cn 𝑆) ∈ V
338, 26, 27xkotf 23609 . . . . . . . 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 6744 . . . . . . . 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 5328 . . . . . 6 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V
37 fiuni 9466 . . . . . 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 6920 . . . . . 6 (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ V
40 unitg 22990 . . . . . 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 2766 . . . 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 2800 . . 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 5105 . . . 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 4034 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ⊆ (𝑅 Cn 𝑆))
4825, 47eqssd 4013 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912   × cxp 5687  ran crn 5690  cima 5692  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  ficfi 9448  t crest 17467  topGenctg 17484  Topctop 22915   Cn ccn 23248  Compccmp 23410  ko cxko 23585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cmp 23411  df-xko 23587
This theorem is referenced by:  xkotopon  23624  xkohaus  23677  xkoptsub  23678
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