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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 23021 | . . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | connsubclo.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
7 | 6 | topopn 22899 | . . . . . . 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 5329 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
11 | connsubclo.5 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐽) | |
12 | elrestr 17443 | . . . . 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 4206 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ∩ 𝐴) = (𝐵 ∩ 𝐴)) | |
17 | 16 | rspceeqv 3630 | . . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴)) → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)) |
18 | 3, 15, 17 | sylancl 584 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)) |
19 | 6 | restcld 23167 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))) |
20 | 5, 9, 19 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))) |
21 | 18, 20 | mpbird 256 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴))) |
22 | 1, 2, 13, 14, 21 | connclo 23410 | . . 3 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = ∪ (𝐽 ↾t 𝐴)) |
23 | 6 | restuni 23157 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
24 | 5, 9, 23 | syl2anc 582 | . . 3 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
25 | 22, 24 | eqtr4d 2769 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴) |
26 | sseqin2 4216 | . 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 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 Vcvv 3462 ∩ cin 3946 ⊆ wss 3947 ∅c0 4325 ∪ cuni 4913 ‘cfv 6554 (class class class)co 7424 ↾t crest 17435 Topctop 22886 Clsdccld 23011 Conncconn 23406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-en 8975 df-fin 8978 df-fi 9454 df-rest 17437 df-topgen 17458 df-top 22887 df-topon 22904 df-bases 22940 df-cld 23014 df-conn 23407 |
This theorem is referenced by: conncn 23421 conncompclo 23430 |
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