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Mirrors > Home > MPE Home > Th. List > llycmpkgen | Structured version Visualization version GIF version |
Description: A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
llycmpkgen | ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen) |
Step | Hyp | Ref | 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) | |
4 | 1 | topopn 21088 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ 𝑛-Locally Comp → ∪ 𝐽 ∈ 𝐽) |
6 | 5 | adantr 474 | . . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ 𝐽) |
7 | simpr 479 | . . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽) | |
8 | nllyi 21656 | . . . 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 1494 | . . 3 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 ⊆ ∪ 𝐽 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) |
10 | simpr 479 | . . . 4 ⊢ ((𝑘 ⊆ ∪ 𝐽 ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝐽 ↾t 𝑘) ∈ Comp) | |
11 | 10 | reximi 3219 | . . 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 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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