MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xkobval Structured version   Visualization version   GIF version

Theorem xkobval 21612
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 6937 . 2 ran 𝑇 = {𝑠 ∣ ∃𝑘𝐾𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}}
3 oveq2 6823 . . . . . 6 (𝑥 = 𝑘 → (𝑅t 𝑥) = (𝑅t 𝑘))
43eleq1d 2825 . . . . 5 (𝑥 = 𝑘 → ((𝑅t 𝑥) ∈ Comp ↔ (𝑅t 𝑘) ∈ Comp))
54rexrab 3512 . . . 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 3283 . . . 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 3231 . . . . 5 (∃𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ↔ ((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
98rexbii 3180 . . . 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 292 . . 3 (∃𝑘𝐾𝑣𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ 𝒫 𝑋𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
1110abbii 2878 . 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 2783 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 1632  wcel 2140  {cab 2747  wrex 3052  {crab 3055  wss 3716  𝒫 cpw 4303   cuni 4589  ran crn 5268  cima 5270  (class class class)co 6815  cmpt2 6817  t crest 16304   Cn ccn 21251  Compccmp 21412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-cnv 5275  df-dm 5277  df-rn 5278  df-iota 6013  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820
This theorem is referenced by:  xkoccn  21645  xkoco1cn  21683  xkoco2cn  21684  xkoinjcn  21713
  Copyright terms: Public domain W3C validator