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Theorem xkouni 22135
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 5938 . . . . . . . . 9 (𝑓 “ ∅) = ∅
2 0ss 4347 . . . . . . . . 9 ∅ ⊆ 𝑆
31, 2eqsstri 3998 . . . . . . . 8 (𝑓 “ ∅) ⊆ 𝑆
43a1i 11 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (𝑓 “ ∅) ⊆ 𝑆)
54ralrimiva 3179 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ 𝑆)
6 rabid2 3379 . . . . . 6 ((𝑅 Cn 𝑆) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ 𝑆} ↔ ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ 𝑆)
75, 6sylibr 235 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ 𝑆})
8 eqid 2818 . . . . . 6 𝑅 = 𝑅
9 simpl 483 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
10 simpr 485 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
11 0ss 4347 . . . . . . 7 ∅ ⊆ 𝑅
1211a1i 11 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∅ ⊆ 𝑅)
13 rest0 21705 . . . . . . . 8 (𝑅 ∈ Top → (𝑅t ∅) = {∅})
1413adantr 481 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅t ∅) = {∅})
15 0cmp 21930 . . . . . . 7 {∅} ∈ Comp
1614, 15syl6eqel 2918 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅t ∅) ∈ Comp)
17 eqid 2818 . . . . . . . 8 𝑆 = 𝑆
1817topopn 21442 . . . . . . 7 (𝑆 ∈ Top → 𝑆𝑆)
1918adantl 482 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆𝑆)
208, 9, 10, 12, 16, 19xkoopn 22125 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ 𝑆} ∈ (𝑆ko 𝑅))
217, 20eqeltrd 2910 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆ko 𝑅))
22 xkouni.1 . . . 4 𝐽 = (𝑆ko 𝑅)
2321, 22eleqtrrdi 2921 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ 𝐽)
24 elssuni 4859 . . 3 ((𝑅 Cn 𝑆) ∈ 𝐽 → (𝑅 Cn 𝑆) ⊆ 𝐽)
2523, 24syl 17 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ⊆ 𝐽)
26 eqid 2818 . . . . . 6 {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}
27 eqid 2818 . . . . . 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 22123 . . . . 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 4840 . . . 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 4839 . . . 4 𝐽 = (𝑆ko 𝑅)
31 ovex 7178 . . . . . . . 8 (𝑅 Cn 𝑆) ∈ V
3231pwex 5272 . . . . . . 7 𝒫 (𝑅 Cn 𝑆) ∈ V
338, 26, 27xkotf 22121 . . . . . . . 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 6513 . . . . . . . 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 5217 . . . . . 6 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V
37 fiuni 8880 . . . . . 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 6676 . . . . . 6 (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ V
40 unitg 21503 . . . . . 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 2844 . . . 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 2878 . . 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 5013 . . . 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 219 . . 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 4002 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ⊆ (𝑅 Cn 𝑆))
4825, 47eqssd 3981 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557   cuni 4830   × cxp 5546  ran crn 5549  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  ficfi 8862  t crest 16682  topGenctg 16699  Topctop 21429   Cn ccn 21760  Compccmp 21922  ko cxko 22097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cmp 21923  df-xko 22099
This theorem is referenced by:  xkotopon  22136  xkohaus  22189  xkoptsub  22190
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