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Mirrors > Home > MPE Home > Th. List > connsubclo | Structured version Visualization version GIF version |
Description: If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
connsubclo.1 | ⊢ 𝑋 = ∪ 𝐽 |
connsubclo.3 | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
connsubclo.4 | ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ Conn) |
connsubclo.5 | ⊢ (𝜑 → 𝐵 ∈ 𝐽) |
connsubclo.6 | ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅) |
connsubclo.7 | ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) |
Ref | Expression |
---|---|
connsubclo | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . 4 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴) | |
2 | connsubclo.4 | . . . 4 ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ Conn) | |
3 | connsubclo.7 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) | |
4 | cldrcl 22885 | . . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | connsubclo.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
7 | 6 | topopn 22763 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
8 | 5, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽) |
9 | connsubclo.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
10 | 8, 9 | ssexd 5317 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
11 | connsubclo.5 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐽) | |
12 | elrestr 17383 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝐵 ∈ 𝐽) → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
13 | 5, 10, 11, 12 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
14 | connsubclo.6 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅) | |
15 | eqid 2726 | . . . . . 6 ⊢ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴) | |
16 | ineq1 4200 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ∩ 𝐴) = (𝐵 ∩ 𝐴)) | |
17 | 16 | rspceeqv 3628 | . . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴)) → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)) |
18 | 3, 15, 17 | sylancl 585 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)) |
19 | 6 | restcld 23031 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))) |
20 | 5, 9, 19 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))) |
21 | 18, 20 | mpbird 257 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴))) |
22 | 1, 2, 13, 14, 21 | connclo 23274 | . . 3 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = ∪ (𝐽 ↾t 𝐴)) |
23 | 6 | restuni 23021 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
24 | 5, 9, 23 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
25 | 22, 24 | eqtr4d 2769 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴) |
26 | sseqin2 4210 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) | |
27 | 25, 26 | sylibr 233 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∃wrex 3064 Vcvv 3468 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 ∪ cuni 4902 ‘cfv 6537 (class class class)co 7405 ↾t crest 17375 Topctop 22750 Clsdccld 22875 Conncconn 23270 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-en 8942 df-fin 8945 df-fi 9408 df-rest 17377 df-topgen 17398 df-top 22751 df-topon 22768 df-bases 22804 df-cld 22878 df-conn 23271 |
This theorem is referenced by: conncn 23285 conncompclo 23294 |
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