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Theorem llycmpkgen 23576
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 2735 . 2 𝐽 = 𝐽
2 nllytop 23497 . 2 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
3 simpl 482 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → 𝐽 ∈ 𝑛-Locally Comp)
41topopn 22928 . . . . . 6 (𝐽 ∈ Top → 𝐽𝐽)
52, 4syl 17 . . . . 5 (𝐽 ∈ 𝑛-Locally Comp → 𝐽𝐽)
65adantr 480 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → 𝐽𝐽)
7 simpr 484 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → 𝑥 𝐽)
8 nllyi 23499 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐽𝐽𝑥 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp))
93, 6, 7, 8syl3anc 1370 . . 3 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp))
10 simpr 484 . . . 4 ((𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp) → (𝐽t 𝑘) ∈ Comp)
1110reximi 3082 . . 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 23574 1 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wrex 3068  wss 3963  {csn 4631   cuni 4912  ran crn 5690  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  neicnei 23121  Compccmp 23410  𝑛-Locally cnlly 23489  𝑘Genckgen 23557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-ntr 23044  df-nei 23122  df-cmp 23411  df-nlly 23491  df-kgen 23558
This theorem is referenced by:  txkgen  23676
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