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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 2737 | . . . 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 23034 | . . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
| 5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) | 
| 6 | connsubclo.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
| 7 | 6 | topopn 22912 | . . . . . . 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 5324 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) | 
| 11 | connsubclo.5 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐽) | |
| 12 | elrestr 17473 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝐵 ∈ 𝐽) → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
| 13 | 5, 10, 11, 12 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | 
| 14 | connsubclo.6 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅) | |
| 15 | eqid 2737 | . . . . . 6 ⊢ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴) | |
| 16 | ineq1 4213 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ∩ 𝐴) = (𝐵 ∩ 𝐴)) | |
| 17 | 16 | rspceeqv 3645 | . . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴)) → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)) | 
| 18 | 3, 15, 17 | sylancl 586 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)) | 
| 19 | 6 | restcld 23180 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))) | 
| 20 | 5, 9, 19 | syl2anc 584 | . . . . 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 23423 | . . 3 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = ∪ (𝐽 ↾t 𝐴)) | 
| 23 | 6 | restuni 23170 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) | 
| 24 | 5, 9, 23 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴)) | 
| 25 | 22, 24 | eqtr4d 2780 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴) | 
| 26 | sseqin2 4223 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) | |
| 27 | 25, 26 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 ‘cfv 6561 (class class class)co 7431 ↾t crest 17465 Topctop 22899 Clsdccld 23024 Conncconn 23419 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-en 8986 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-cld 23027 df-conn 23420 | 
| This theorem is referenced by: conncn 23434 conncompclo 23443 | 
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