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Theorem xkoopn 23612
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} ∈ (𝑆ko 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 7463 . . . . . . 7 (𝑅 Cn 𝑆) ∈ V
21pwex 5385 . . . . . 6 𝒫 (𝑅 Cn 𝑆) ∈ V
3 xkoopn.x . . . . . . . 8 𝑋 = 𝑅
4 eqid 2734 . . . . . . . 8 {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}
5 eqid 2734 . . . . . . . 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 23608 . . . . . . 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 6743 . . . . . . 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 5327 . . . . 5 ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V
10 ssfii 9456 . . . . 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 6919 . . . . 5 (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})) ∈ V
13 bastg 22988 . . . . 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 4004 . . 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 oveq2 7438 . . . . . . 7 (𝑥 = 𝐴 → (𝑅t 𝑥) = (𝑅t 𝐴))
1716eleq1d 2823 . . . . . 6 (𝑥 = 𝐴 → ((𝑅t 𝑥) ∈ Comp ↔ (𝑅t 𝐴) ∈ Comp))
18 xkoopn.a . . . . . . 7 (𝜑𝐴𝑋)
19 xkoopn.r . . . . . . . 8 (𝜑𝑅 ∈ Top)
203topopn 22927 . . . . . . . 8 (𝑅 ∈ Top → 𝑋𝑅)
21 elpw2g 5338 . . . . . . . 8 (𝑋𝑅 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
2219, 20, 213syl 18 . . . . . . 7 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
2318, 22mpbird 257 . . . . . 6 (𝜑𝐴 ∈ 𝒫 𝑋)
24 xkoopn.c . . . . . 6 (𝜑 → (𝑅t 𝐴) ∈ Comp)
2517, 23, 24elrabd 3696 . . . . 5 (𝜑𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp})
26 xkoopn.u . . . . 5 (𝜑𝑈𝑆)
27 eqidd 2735 . . . . 5 (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈})
28 imaeq2 6075 . . . . . . . . 9 (𝑘 = 𝐴 → (𝑓𝑘) = (𝑓𝐴))
2928sseq1d 4026 . . . . . . . 8 (𝑘 = 𝐴 → ((𝑓𝑘) ⊆ 𝑣 ↔ (𝑓𝐴) ⊆ 𝑣))
3029rabbidv 3440 . . . . . . 7 (𝑘 = 𝐴 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑣})
3130eqeq2d 2745 . . . . . 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}))
32 sseq2 4021 . . . . . . . 8 (𝑣 = 𝑈 → ((𝑓𝐴) ⊆ 𝑣 ↔ (𝑓𝐴) ⊆ 𝑈))
3332rabbidv 3440 . . . . . . 7 (𝑣 = 𝑈 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈})
3433eqeq2d 2745 . . . . . 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}))
3531, 34rspc2ev 3634 . . . . 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})
3625, 26, 27, 35syl3anc 1370 . . . 4 (𝜑 → ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}∃𝑣𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
371rabex 5344 . . . . 5 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ V
38 eqeq1 2738 . . . . . 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}))
39382rexbidv 3219 . . . . 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}))
405rnmpo 7565 . . . . 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}}
4137, 39, 40elab2 3684 . . . 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})
4236, 41sylibr 234 . . 3 (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
4315, 42sselid 3992 . 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}))))
44 xkoopn.s . . 3 (𝜑𝑆 ∈ Top)
453, 4, 5xkoval 23610 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
4619, 44, 45syl2anc 584 . 2 (𝜑 → (𝑆ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
4743, 46eleqtrrd 2841 1 (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ (𝑆ko 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536  wcel 2105  wrex 3067  {crab 3432  Vcvv 3477  wss 3962  𝒫 cpw 4604   cuni 4911   × cxp 5686  ran crn 5689  cima 5691  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  ficfi 9447  t crest 17466  topGenctg 17483  Topctop 22914   Cn ccn 23247  Compccmp 23409  ko cxko 23584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-1o 8504  df-en 8984  df-fin 8987  df-fi 9448  df-topgen 17489  df-top 22915  df-xko 23586
This theorem is referenced by:  xkouni  23622  xkohaus  23676  xkoptsub  23677  xkoco1cn  23680  xkoco2cn  23681  xkococnlem  23682
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