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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 2728 | . . . 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 22958 | . . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | connsubclo.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
7 | 6 | topopn 22836 | . . . . . . 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 5328 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
11 | connsubclo.5 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐽) | |
12 | elrestr 17419 | . . . . 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 2728 | . . . . . 6 ⊢ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴) | |
16 | ineq1 4207 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ∩ 𝐴) = (𝐵 ∩ 𝐴)) | |
17 | 16 | rspceeqv 3633 | . . . . . 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 23104 | . . . . . 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 23347 | . . 3 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = ∪ (𝐽 ↾t 𝐴)) |
23 | 6 | restuni 23094 | . . . 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 2771 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴) |
26 | sseqin2 4217 | . 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 2937 ∃wrex 3067 Vcvv 3473 ∩ cin 3948 ⊆ wss 3949 ∅c0 4326 ∪ cuni 4912 ‘cfv 6553 (class class class)co 7426 ↾t crest 17411 Topctop 22823 Clsdccld 22948 Conncconn 23343 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 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 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-en 8973 df-fin 8976 df-fi 9444 df-rest 17413 df-topgen 17434 df-top 22824 df-topon 22841 df-bases 22877 df-cld 22951 df-conn 23344 |
This theorem is referenced by: conncn 23358 conncompclo 23367 |
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