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

Theorem llycmpkgen 21265
 Description: A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
llycmpkgen (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen)

Proof of Theorem llycmpkgen
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . 2 𝐽 = 𝐽
2 nllytop 21186 . 2 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
3 simpl 473 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → 𝐽 ∈ 𝑛-Locally Comp)
41topopn 20636 . . . . . 6 (𝐽 ∈ Top → 𝐽𝐽)
52, 4syl 17 . . . . 5 (𝐽 ∈ 𝑛-Locally Comp → 𝐽𝐽)
65adantr 481 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → 𝐽𝐽)
7 simpr 477 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → 𝑥 𝐽)
8 nllyi 21188 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐽𝐽𝑥 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp))
93, 6, 7, 8syl3anc 1323 . . 3 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp))
10 simpr 477 . . . 4 ((𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp) → (𝐽t 𝑘) ∈ Comp)
1110reximi 3005 . . 3 (∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
129, 11syl 17 . 2 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
131, 2, 12llycmpkgen2 21263 1 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1987  ∃wrex 2908   ⊆ wss 3555  {csn 4148  ∪ cuni 4402  ran crn 5075  ‘cfv 5847  (class class class)co 6604   ↾t crest 16002  Topctop 20617  neicnei 20811  Compccmp 21099  𝑛-Locally cnlly 21178  𝑘Genckgen 21246 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-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 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-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-ntr 20734  df-nei 20812  df-cmp 21100  df-nlly 21180  df-kgen 21247 This theorem is referenced by:  txkgen  21365
 Copyright terms: Public domain W3C validator