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Theorem llycmpkgen 21733
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 2825 . 2 𝐽 = 𝐽
2 nllytop 21654 . 2 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
3 simpl 476 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → 𝐽 ∈ 𝑛-Locally Comp)
41topopn 21088 . . . . . 6 (𝐽 ∈ Top → 𝐽𝐽)
52, 4syl 17 . . . . 5 (𝐽 ∈ 𝑛-Locally Comp → 𝐽𝐽)
65adantr 474 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → 𝐽𝐽)
7 simpr 479 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → 𝑥 𝐽)
8 nllyi 21656 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐽𝐽𝑥 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp))
93, 6, 7, 8syl3anc 1494 . . 3 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp))
10 simpr 479 . . . 4 ((𝑘 𝐽 ∧ (𝐽t 𝑘) ∈ Comp) → (𝐽t 𝑘) ∈ Comp)
1110reximi 3219 . . 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 21731 1 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2164  wrex 3118  wss 3798  {csn 4399   cuni 4660  ran crn 5347  cfv 6127  (class class class)co 6910  t crest 16441  Topctop 21075  neicnei 21279  Compccmp 21567  𝑛-Locally cnlly 21646  𝑘Genckgen 21714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-oadd 7835  df-er 8014  df-en 8229  df-fin 8232  df-fi 8592  df-rest 16443  df-topgen 16464  df-top 21076  df-topon 21093  df-bases 21128  df-ntr 21202  df-nei 21280  df-cmp 21568  df-nlly 21648  df-kgen 21715
This theorem is referenced by:  txkgen  21833
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