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Theorem llycmpkgen 23561
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 2736 . 2 𝐽 = 𝐽
2 nllytop 23482 . 2 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
3 simpl 482 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → 𝐽 ∈ 𝑛-Locally Comp)
41topopn 22913 . . . . . 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 23484 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐽𝐽𝑥 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp))
93, 6, 7, 8syl3anc 1372 . . 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 3083 . . 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 23559 1 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2107  wrex 3069  wss 3950  {csn 4625   cuni 4906  ran crn 5685  cfv 6560  (class class class)co 7432  t crest 17466  Topctop 22900  neicnei 23106  Compccmp 23395  𝑛-Locally cnlly 23474  𝑘Genckgen 23542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-en 8987  df-fin 8990  df-fi 9452  df-rest 17468  df-topgen 17489  df-top 22901  df-topon 22918  df-bases 22954  df-ntr 23029  df-nei 23107  df-cmp 23396  df-nlly 23476  df-kgen 23543
This theorem is referenced by:  txkgen  23661
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