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Theorem xkoopn 21440
Description: A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
xkoopn.x 𝑋 = 𝑅
xkoopn.r (𝜑𝑅 ∈ Top)
xkoopn.s (𝜑𝑆 ∈ Top)
xkoopn.a (𝜑𝐴𝑋)
xkoopn.c (𝜑 → (𝑅t 𝐴) ∈ Comp)
xkoopn.u (𝜑𝑈𝑆)
Assertion
Ref Expression
xkoopn (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ (𝑆 ^ko 𝑅))
Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑆,𝑓   𝑈,𝑓
Allowed substitution hints:   𝜑(𝑓)   𝑋(𝑓)

Proof of Theorem xkoopn
Dummy variables 𝑘 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6718 . . . . . . 7 (𝑅 Cn 𝑆) ∈ V
21pwex 4878 . . . . . 6 𝒫 (𝑅 Cn 𝑆) ∈ V
3 xkoopn.x . . . . . . . 8 𝑋 = 𝑅
4 eqid 2651 . . . . . . . 8 {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}
5 eqid 2651 . . . . . . . 8 (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
63, 4, 5xkotf 21436 . . . . . . 7 (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}):({𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
7 frn 6091 . . . . . . 7 ((𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}):({𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆))
86, 7ax-mp 5 . . . . . 6 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆)
92, 8ssexi 4836 . . . . 5 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V
10 ssfii 8366 . . . . 5 (ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})))
119, 10ax-mp 5 . . . 4 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
12 fvex 6239 . . . . 5 (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ V
13 bastg 20818 . . . . 5 ((fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ V → (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ⊆ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
1412, 13ax-mp 5 . . . 4 (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ⊆ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})))
1511, 14sstri 3645 . . 3 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})))
16 xkoopn.a . . . . . . 7 (𝜑𝐴𝑋)
17 xkoopn.r . . . . . . . 8 (𝜑𝑅 ∈ Top)
183topopn 20759 . . . . . . . 8 (𝑅 ∈ Top → 𝑋𝑅)
19 elpw2g 4857 . . . . . . . 8 (𝑋𝑅 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
2017, 18, 193syl 18 . . . . . . 7 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
2116, 20mpbird 247 . . . . . 6 (𝜑𝐴 ∈ 𝒫 𝑋)
22 xkoopn.c . . . . . 6 (𝜑 → (𝑅t 𝐴) ∈ Comp)
23 oveq2 6698 . . . . . . . 8 (𝑥 = 𝐴 → (𝑅t 𝑥) = (𝑅t 𝐴))
2423eleq1d 2715 . . . . . . 7 (𝑥 = 𝐴 → ((𝑅t 𝑥) ∈ Comp ↔ (𝑅t 𝐴) ∈ Comp))
2524elrab 3396 . . . . . 6 (𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ (𝑅t 𝐴) ∈ Comp))
2621, 22, 25sylanbrc 699 . . . . 5 (𝜑𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp})
27 xkoopn.u . . . . 5 (𝜑𝑈𝑆)
28 eqidd 2652 . . . . 5 (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈})
29 imaeq2 5497 . . . . . . . . 9 (𝑘 = 𝐴 → (𝑓𝑘) = (𝑓𝐴))
3029sseq1d 3665 . . . . . . . 8 (𝑘 = 𝐴 → ((𝑓𝑘) ⊆ 𝑣 ↔ (𝑓𝐴) ⊆ 𝑣))
3130rabbidv 3220 . . . . . . 7 (𝑘 = 𝐴 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑣})
3231eqeq2d 2661 . . . . . 6 (𝑘 = 𝐴 → ({𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑣}))
33 sseq2 3660 . . . . . . . 8 (𝑣 = 𝑈 → ((𝑓𝐴) ⊆ 𝑣 ↔ (𝑓𝐴) ⊆ 𝑈))
3433rabbidv 3220 . . . . . . 7 (𝑣 = 𝑈 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈})
3534eqeq2d 2661 . . . . . 6 (𝑣 = 𝑈 → ({𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑣} ↔ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈}))
3632, 35rspc2ev 3355 . . . . 5 ((𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp} ∧ 𝑈𝑆 ∧ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈}) → ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}∃𝑣𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
3726, 27, 28, 36syl3anc 1366 . . . 4 (𝜑 → ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}∃𝑣𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
381rabex 4845 . . . . 5 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ V
39 eqeq1 2655 . . . . . 6 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} → (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
40392rexbidv 3086 . . . . 5 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} → (∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}∃𝑣𝑆 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}∃𝑣𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
415rnmpt2 6812 . . . . 5 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = {𝑦 ∣ ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}∃𝑣𝑆 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}}
4238, 40, 41elab2 3386 . . . 4 ({𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}∃𝑣𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
4337, 42sylibr 224 . . 3 (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
4415, 43sseldi 3634 . 2 (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
45 xkoopn.s . . 3 (𝜑𝑆 ∈ Top)
463, 4, 5xkoval 21438 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
4717, 45, 46syl2anc 694 . 2 (𝜑 → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
4844, 47eleqtrrd 2733 1 (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ (𝑆 ^ko 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191   cuni 4468   × cxp 5141  ran crn 5144  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  ficfi 8357  t crest 16128  topGenctg 16145  Topctop 20746   Cn ccn 21076  Compccmp 21237   ^ko cxko 21412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-1o 7605  df-en 7998  df-fin 8001  df-fi 8358  df-topgen 16151  df-top 20747  df-xko 21414
This theorem is referenced by:  xkouni  21450  xkohaus  21504  xkoptsub  21505  xkoco1cn  21508  xkoco2cn  21509  xkococnlem  21510
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