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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 2818 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | cmptop 21931 | . 2 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | |
3 | 2 | adantr 481 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Top) |
4 | 1 | topopn 21442 | . . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 3, 4 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ 𝐽) |
6 | simpr 485 | . . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽) | |
7 | 6 | snssd 4734 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ⊆ ∪ 𝐽) |
8 | opnneiss 21654 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝐽 ∈ 𝐽 ∧ {𝑥} ⊆ ∪ 𝐽) → ∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥})) | |
9 | 3, 5, 7, 8 | syl3anc 1363 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥})) |
10 | 1 | restid 16695 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∪ 𝐽) = 𝐽) |
11 | 3, 10 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → (𝐽 ↾t ∪ 𝐽) = 𝐽) |
12 | simpl 483 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Comp) | |
13 | 11, 12 | eqeltrd 2910 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → (𝐽 ↾t ∪ 𝐽) ∈ Comp) |
14 | oveq2 7153 | . . . . 5 ⊢ (𝑘 = ∪ 𝐽 → (𝐽 ↾t 𝑘) = (𝐽 ↾t ∪ 𝐽)) | |
15 | 14 | eleq1d 2894 | . . . 4 ⊢ (𝑘 = ∪ 𝐽 → ((𝐽 ↾t 𝑘) ∈ Comp ↔ (𝐽 ↾t ∪ 𝐽) ∈ Comp)) |
16 | 15 | rspcev 3620 | . . 3 ⊢ ((∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t ∪ 𝐽) ∈ Comp) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp) |
17 | 9, 13, 16 | syl2anc 584 | . 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp) |
18 | 1, 2, 17 | llycmpkgen2 22086 | 1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ ran 𝑘Gen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ⊆ wss 3933 {csn 4557 ∪ cuni 4830 ran crn 5549 ‘cfv 6348 (class class class)co 7145 ↾t crest 16682 Topctop 21429 neicnei 21633 Compccmp 21922 𝑘Genckgen 22069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-oadd 8095 df-er 8278 df-en 8498 df-fin 8501 df-fi 8863 df-rest 16684 df-topgen 16705 df-top 21430 df-topon 21447 df-bases 21482 df-ntr 21556 df-nei 21634 df-cmp 21923 df-kgen 22070 |
This theorem is referenced by: (None) |
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