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Theorem xkobval 23480
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})
21rnmpo 7525 . 2 ran 𝑇 = {𝑠 ∣ ∃𝑘𝐾𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}}
3 oveq2 7398 . . . . . 6 (𝑥 = 𝑘 → (𝑅t 𝑥) = (𝑅t 𝑘))
43eleq1d 2814 . . . . 5 (𝑥 = 𝑘 → ((𝑅t 𝑥) ∈ Comp ↔ (𝑅t 𝑘) ∈ Comp))
54rexrab 3670 . . . 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 3300 . . . 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 3170 . . . . 5 (∃𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ↔ ((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
98rexbii 3077 . . . 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 303 . . 3 (∃𝑘𝐾𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ 𝒫 𝑋𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
1110abbii 2797 . 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 2753 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 395   = wceq 1540  wcel 2109  {cab 2708  wrex 3054  {crab 3408  wss 3917  𝒫 cpw 4566   cuni 4874  ran crn 5642  cima 5644  (class class class)co 7390  cmpo 7392  t crest 17390   Cn ccn 23118  Compccmp 23280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-cnv 5649  df-dm 5651  df-rn 5652  df-iota 6467  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395
This theorem is referenced by:  xkoccn  23513  xkoco1cn  23551  xkoco2cn  23552  xkoinjcn  23581
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