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Theorem xkobval 21199
 Description: Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypotheses
Ref Expression
xkoval.x 𝑋 = 𝑅
xkoval.k 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}
xkoval.t 𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
Assertion
Ref Expression
xkobval ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})}
Distinct variable groups:   𝑘,𝑠,𝑣,𝐾   𝑓,𝑘,𝑠,𝑣,𝑥,𝑅   𝑆,𝑓,𝑘,𝑠,𝑣,𝑥   𝑇,𝑠   𝑘,𝑋,𝑥
Allowed substitution hints:   𝑇(𝑥,𝑣,𝑓,𝑘)   𝐾(𝑥,𝑓)   𝑋(𝑣,𝑓,𝑠)

Proof of Theorem xkobval
StepHypRef Expression
1 xkoval.t . . 3 𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
21rnmpt2 6668 . 2 ran 𝑇 = {𝑠 ∣ ∃𝑘𝐾𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}}
3 oveq2 6557 . . . . . 6 (𝑥 = 𝑘 → (𝑅t 𝑥) = (𝑅t 𝑘))
43eleq1d 2672 . . . . 5 (𝑥 = 𝑘 → ((𝑅t 𝑥) ∈ Comp ↔ (𝑅t 𝑘) ∈ Comp))
54rexrab 3337 . . . 4 (∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}∃𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ 𝒫 𝑋((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
6 xkoval.k . . . . 5 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}
76rexeqi 3120 . . . 4 (∃𝑘𝐾𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}∃𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
8 r19.42v 3073 . . . . 5 (∃𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ↔ ((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
98rexbii 3023 . . . 4 (∃𝑘 ∈ 𝒫 𝑋𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 𝑋((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
105, 7, 93bitr4i 291 . . 3 (∃𝑘𝐾𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ 𝒫 𝑋𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
1110abbii 2726 . 2 {𝑠 ∣ ∃𝑘𝐾𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})}
122, 11eqtri 2632 1 ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ran crn 5039   “ cima 5041  (class class class)co 6549   ↦ cmpt2 6551   ↾t crest 15904   Cn ccn 20838  Compccmp 20999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  xkoccn  21232  xkoco1cn  21270  xkoco2cn  21271  xkoinjcn  21300
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