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Theorem xkoopn 21305
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 6635 . . . . . . 7 (𝑅 Cn 𝑆) ∈ V
21pwex 4810 . . . . . 6 𝒫 (𝑅 Cn 𝑆) ∈ V
3 xkoopn.x . . . . . . . 8 𝑋 = 𝑅
4 eqid 2621 . . . . . . . 8 {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}
5 eqid 2621 . . . . . . . 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 21301 . . . . . . 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 6012 . . . . . . 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 4765 . . . . 5 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V
10 ssfii 8272 . . . . 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 6160 . . . . 5 (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ V
13 bastg 20684 . . . . 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 3593 . . 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 20633 . . . . . . . 8 (𝑅 ∈ Top → 𝑋𝑅)
19 elpw2g 4789 . . . . . . . 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 6615 . . . . . . . 8 (𝑥 = 𝐴 → (𝑅t 𝑥) = (𝑅t 𝐴))
2423eleq1d 2683 . . . . . . 7 (𝑥 = 𝐴 → ((𝑅t 𝑥) ∈ Comp ↔ (𝑅t 𝐴) ∈ Comp))
2524elrab 3347 . . . . . 6 (𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ (𝑅t 𝐴) ∈ Comp))
2621, 22, 25sylanbrc 697 . . . . 5 (𝜑𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp})
27 xkoopn.u . . . . 5 (𝜑𝑈𝑆)
28 eqidd 2622 . . . . 5 (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈})
29 imaeq2 5423 . . . . . . . . 9 (𝑘 = 𝐴 → (𝑓𝑘) = (𝑓𝐴))
3029sseq1d 3613 . . . . . . . 8 (𝑘 = 𝐴 → ((𝑓𝑘) ⊆ 𝑣 ↔ (𝑓𝐴) ⊆ 𝑣))
3130rabbidv 3177 . . . . . . 7 (𝑘 = 𝐴 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑣})
3231eqeq2d 2631 . . . . . 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 3608 . . . . . . . 8 (𝑣 = 𝑈 → ((𝑓𝐴) ⊆ 𝑣 ↔ (𝑓𝐴) ⊆ 𝑈))
3433rabbidv 3177 . . . . . . 7 (𝑣 = 𝑈 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈})
3534eqeq2d 2631 . . . . . 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 3309 . . . . 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 1323 . . . 4 (𝜑 → ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}∃𝑣𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
381rabex 4775 . . . . 5 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ V
39 eqeq1 2625 . . . . . 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 3050 . . . . 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 6726 . . . . 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 3338 . . . 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 3582 . 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 21303 . . 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 692 . 2 (𝜑 → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
4844, 47eleqtrrd 2701 1 (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ (𝑆 ^ko 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  wrex 2908  {crab 2911  Vcvv 3186  wss 3556  𝒫 cpw 4132   cuni 4404   × cxp 5074  ran crn 5077  cima 5079  wf 5845  cfv 5849  (class class class)co 6607  cmpt2 6609  ficfi 8263  t crest 16005  topGenctg 16022  Topctop 20620   Cn ccn 20941  Compccmp 21102   ^ko cxko 21277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-1o 7508  df-en 7903  df-fin 7906  df-fi 8264  df-topgen 16028  df-top 20621  df-xko 21279
This theorem is referenced by:  xkouni  21315  xkohaus  21369  xkoptsub  21370  xkoco1cn  21373  xkoco2cn  21374  xkococnlem  21375
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