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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 2771 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | nllytop 21497 | . 2 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top) | |
3 | simpl 468 | . . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ 𝑛-Locally Comp) | |
4 | 1 | topopn 20931 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ 𝑛-Locally Comp → ∪ 𝐽 ∈ 𝐽) |
6 | 5 | adantr 466 | . . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ 𝐽) |
7 | simpr 471 | . . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽) | |
8 | nllyi 21499 | . . . 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 1476 | . . 3 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 ⊆ ∪ 𝐽 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) |
10 | simpr 471 | . . . 4 ⊢ ((𝑘 ⊆ ∪ 𝐽 ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝐽 ↾t 𝑘) ∈ Comp) | |
11 | 10 | reximi 3159 | . . 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 21574 | 1 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 ∃wrex 3062 ⊆ wss 3723 {csn 4316 ∪ cuni 4574 ran crn 5250 ‘cfv 6031 (class class class)co 6793 ↾t crest 16289 Topctop 20918 neicnei 21122 Compccmp 21410 𝑛-Locally cnlly 21489 𝑘Genckgen 21557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-oadd 7717 df-er 7896 df-en 8110 df-fin 8113 df-fi 8473 df-rest 16291 df-topgen 16312 df-top 20919 df-topon 20936 df-bases 20971 df-ntr 21045 df-nei 21123 df-cmp 21411 df-nlly 21491 df-kgen 21558 |
This theorem is referenced by: txkgen 21676 |
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