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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 2725 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | cmptop 23343 | . 2 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | |
3 | 2 | adantr 479 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Top) |
4 | 1 | topopn 22852 | . . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 3, 4 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ 𝐽) |
6 | simpr 483 | . . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽) | |
7 | 6 | snssd 4814 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ⊆ ∪ 𝐽) |
8 | opnneiss 23066 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝐽 ∈ 𝐽 ∧ {𝑥} ⊆ ∪ 𝐽) → ∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥})) | |
9 | 3, 5, 7, 8 | syl3anc 1368 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥})) |
10 | 1 | restid 17418 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∪ 𝐽) = 𝐽) |
11 | 3, 10 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → (𝐽 ↾t ∪ 𝐽) = 𝐽) |
12 | simpl 481 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Comp) | |
13 | 11, 12 | eqeltrd 2825 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → (𝐽 ↾t ∪ 𝐽) ∈ Comp) |
14 | oveq2 7427 | . . . . 5 ⊢ (𝑘 = ∪ 𝐽 → (𝐽 ↾t 𝑘) = (𝐽 ↾t ∪ 𝐽)) | |
15 | 14 | eleq1d 2810 | . . . 4 ⊢ (𝑘 = ∪ 𝐽 → ((𝐽 ↾t 𝑘) ∈ Comp ↔ (𝐽 ↾t ∪ 𝐽) ∈ Comp)) |
16 | 15 | rspcev 3606 | . . 3 ⊢ ((∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t ∪ 𝐽) ∈ Comp) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp) |
17 | 9, 13, 16 | syl2anc 582 | . 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp) |
18 | 1, 2, 17 | llycmpkgen2 23498 | 1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ ran 𝑘Gen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 ⊆ wss 3944 {csn 4630 ∪ cuni 4909 ran crn 5679 ‘cfv 6549 (class class class)co 7419 ↾t crest 17405 Topctop 22839 neicnei 23045 Compccmp 23334 𝑘Genckgen 23481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-en 8965 df-fin 8968 df-fi 9436 df-rest 17407 df-topgen 17428 df-top 22840 df-topon 22857 df-bases 22893 df-ntr 22968 df-nei 23046 df-cmp 23335 df-kgen 23482 |
This theorem is referenced by: (None) |
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