Theorem llycmpkgen 22253

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.

Step	Hyp	Ref	Expression
1		eqid 2759	. 2 ⊢ ∪ 𝐽 = ∪ 𝐽
2		nllytop 22174	. 2 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
3		simpl 487	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ 𝑛-Locally Comp)
4	1	topopn 21607	. . . . . 6 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
5	2, 4	syl 17	. . . . 5 ⊢ (𝐽 ∈ 𝑛-Locally Comp → ∪ 𝐽 ∈ 𝐽)
6	5	adantr 485	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ 𝐽)
7		simpr 489	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽)
8		nllyi 22176	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ ∪ 𝐽 ∈ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 ⊆ ∪ 𝐽 ∧ (𝐽 ↾t 𝑘) ∈ Comp))
9	3, 6, 7, 8	syl3anc 1369	. . 3 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 ⊆ ∪ 𝐽 ∧ (𝐽 ↾t 𝑘) ∈ Comp))
10		simpr 489	. . . 4 ⊢ ((𝑘 ⊆ ∪ 𝐽 ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝐽 ↾t 𝑘) ∈ Comp)
11	10	reximi 3172	. . 3 ⊢ (∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 ⊆ ∪ 𝐽 ∧ (𝐽 ↾t 𝑘) ∈ Comp) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp)
12	9, 11	syl 17	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp)
13	1, 2, 12	llycmpkgen2 22251	1 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2112  ∃wrex 3072   ⊆ wss 3859  {csn 4523  ∪ cuni 4799  ran crn 5526  ‘cfv 6336  (class class class)co 7151   ↾_t crest 16753  Topctop 21594  neicnei 21798  Compccmp 22087  𝑛-Locally cnlly 22166  𝑘Genckgen 22234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-en 8529  df-fin 8532  df-fi 8909  df-rest 16755  df-topgen 16776  df-top 21595  df-topon 21612  df-bases 21647  df-ntr 21721  df-nei 21799  df-cmp 22088  df-nlly 22168  df-kgen 22235

This theorem is referenced by:  txkgen  22353