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Mirrors > Home > MPE Home > Th. List > cmpkgen | Structured version Visualization version GIF version |
Description: A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
cmpkgen | ⊢ (𝐽 ∈ Comp → 𝐽 ∈ ran 𝑘Gen) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2799 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | cmptop 21527 | . 2 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | |
3 | 2 | adantr 473 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Top) |
4 | 1 | topopn 21039 | . . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 3, 4 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ 𝐽) |
6 | simpr 478 | . . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽) | |
7 | 6 | snssd 4528 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ⊆ ∪ 𝐽) |
8 | opnneiss 21251 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝐽 ∈ 𝐽 ∧ {𝑥} ⊆ ∪ 𝐽) → ∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥})) | |
9 | 3, 5, 7, 8 | syl3anc 1491 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥})) |
10 | 1 | restid 16409 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∪ 𝐽) = 𝐽) |
11 | 3, 10 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → (𝐽 ↾t ∪ 𝐽) = 𝐽) |
12 | simpl 475 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Comp) | |
13 | 11, 12 | eqeltrd 2878 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → (𝐽 ↾t ∪ 𝐽) ∈ Comp) |
14 | oveq2 6886 | . . . . 5 ⊢ (𝑘 = ∪ 𝐽 → (𝐽 ↾t 𝑘) = (𝐽 ↾t ∪ 𝐽)) | |
15 | 14 | eleq1d 2863 | . . . 4 ⊢ (𝑘 = ∪ 𝐽 → ((𝐽 ↾t 𝑘) ∈ Comp ↔ (𝐽 ↾t ∪ 𝐽) ∈ Comp)) |
16 | 15 | rspcev 3497 | . . 3 ⊢ ((∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t ∪ 𝐽) ∈ Comp) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp) |
17 | 9, 13, 16 | syl2anc 580 | . 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp) |
18 | 1, 2, 17 | llycmpkgen2 21682 | 1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ ran 𝑘Gen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 ⊆ wss 3769 {csn 4368 ∪ cuni 4628 ran crn 5313 ‘cfv 6101 (class class class)co 6878 ↾t crest 16396 Topctop 21026 neicnei 21230 Compccmp 21518 𝑘Genckgen 21665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-oadd 7803 df-er 7982 df-en 8196 df-fin 8199 df-fi 8559 df-rest 16398 df-topgen 16419 df-top 21027 df-topon 21044 df-bases 21079 df-ntr 21153 df-nei 21231 df-cmp 21519 df-kgen 21666 |
This theorem is referenced by: (None) |
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