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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 2610 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | nllytop 21086 | . 2 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top) | |
3 | simpl 472 | . . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ 𝑛-Locally Comp) | |
4 | 1 | topopn 20536 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ 𝑛-Locally Comp → ∪ 𝐽 ∈ 𝐽) |
6 | 5 | adantr 480 | . . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ 𝐽) |
7 | simpr 476 | . . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽) | |
8 | nllyi 21088 | . . . 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 1318 | . . 3 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝑘 ⊆ ∪ 𝐽 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) |
10 | simpr 476 | . . . 4 ⊢ ((𝑘 ⊆ ∪ 𝐽 ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝐽 ↾t 𝑘) ∈ Comp) | |
11 | 10 | reximi 2994 | . . 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 21163 | 1 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 {csn 4125 ∪ cuni 4372 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ↾t crest 15904 Topctop 20517 neicnei 20711 Compccmp 20999 𝑛-Locally cnlly 21078 𝑘Genckgen 21146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-er 7629 df-en 7842 df-fin 7845 df-fi 8200 df-rest 15906 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-ntr 20634 df-nei 20712 df-cmp 21000 df-nlly 21080 df-kgen 21147 |
This theorem is referenced by: txkgen 21265 |
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