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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cmpfiiin | Structured version Visualization version GIF version | ||
| Description: In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| cmpfiiin.x | ⊢ 𝑋 = ∪ 𝐽 | 
| cmpfiiin.j | ⊢ (𝜑 → 𝐽 ∈ Comp) | 
| cmpfiiin.s | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑆 ∈ (Clsd‘𝐽)) | 
| cmpfiiin.z | ⊢ ((𝜑 ∧ (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) → (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆) ≠ ∅) | 
| Ref | Expression | 
|---|---|
| cmpfiiin | ⊢ (𝜑 → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) ≠ ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cmpfiiin.j | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Comp) | |
| 2 | cmptop 23404 | . . . . 5 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | |
| 3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) | 
| 4 | cmpfiiin.x | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
| 5 | 4 | topcld 23044 | . . . 4 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) | 
| 6 | 3, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Clsd‘𝐽)) | 
| 7 | cmpfiiin.s | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑆 ∈ (Clsd‘𝐽)) | |
| 8 | 4 | cldss 23038 | . . . . 5 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋) | 
| 9 | 7, 8 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑆 ⊆ 𝑋) | 
| 10 | 9 | ralrimiva 3145 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) | 
| 11 | riinint 5981 | . . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) | |
| 12 | 6, 10, 11 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) | 
| 13 | 6 | snssd 4808 | . . . 4 ⊢ (𝜑 → {𝑋} ⊆ (Clsd‘𝐽)) | 
| 14 | 7 | fmpttd 7134 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐼 ↦ 𝑆):𝐼⟶(Clsd‘𝐽)) | 
| 15 | 14 | frnd 6743 | . . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝐼 ↦ 𝑆) ⊆ (Clsd‘𝐽)) | 
| 16 | 13, 15 | unssd 4191 | . . 3 ⊢ (𝜑 → ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) ⊆ (Clsd‘𝐽)) | 
| 17 | elin 3966 | . . . . . . 7 ⊢ (𝑙 ∈ (𝒫 𝐼 ∩ Fin) ↔ (𝑙 ∈ 𝒫 𝐼 ∧ 𝑙 ∈ Fin)) | |
| 18 | elpwi 4606 | . . . . . . . 8 ⊢ (𝑙 ∈ 𝒫 𝐼 → 𝑙 ⊆ 𝐼) | |
| 19 | 18 | anim1i 615 | . . . . . . 7 ⊢ ((𝑙 ∈ 𝒫 𝐼 ∧ 𝑙 ∈ Fin) → (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) | 
| 20 | 17, 19 | sylbi 217 | . . . . . 6 ⊢ (𝑙 ∈ (𝒫 𝐼 ∩ Fin) → (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) | 
| 21 | cmpfiiin.z | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) → (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆) ≠ ∅) | |
| 22 | nesym 2996 | . . . . . . 7 ⊢ ((𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆) ≠ ∅ ↔ ¬ ∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆)) | |
| 23 | 21, 22 | sylib 218 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) → ¬ ∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆)) | 
| 24 | 20, 23 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝒫 𝐼 ∩ Fin)) → ¬ ∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆)) | 
| 25 | 24 | nrexdv 3148 | . . . 4 ⊢ (𝜑 → ¬ ∃𝑙 ∈ (𝒫 𝐼 ∩ Fin)∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆)) | 
| 26 | elrfirn2 42712 | . . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (∅ ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) ↔ ∃𝑙 ∈ (𝒫 𝐼 ∩ Fin)∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆))) | |
| 27 | 6, 10, 26 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (∅ ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) ↔ ∃𝑙 ∈ (𝒫 𝐼 ∩ Fin)∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆))) | 
| 28 | 25, 27 | mtbird 325 | . . 3 ⊢ (𝜑 → ¬ ∅ ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))) | 
| 29 | cmpfii 23418 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))) → ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) ≠ ∅) | |
| 30 | 1, 16, 28, 29 | syl3anc 1372 | . 2 ⊢ (𝜑 → ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) ≠ ∅) | 
| 31 | 12, 30 | eqnetrd 3007 | 1 ⊢ (𝜑 → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) ≠ ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ∪ cun 3948 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 𝒫 cpw 4599 {csn 4625 ∪ cuni 4906 ∩ cint 4945 ∩ ciin 4991 ↦ cmpt 5224 ran crn 5685 ‘cfv 6560 Fincfn 8986 ficfi 9451 Topctop 22900 Clsdccld 23025 Compccmp 23395 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-om 7889 df-1o 8507 df-en 8987 df-dom 8988 df-fin 8990 df-fi 9452 df-top 22901 df-cld 23028 df-cmp 23396 | 
| This theorem is referenced by: kelac1 43080 | 
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