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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 2778 | . . . 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 21249 | . . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | connsubclo.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
7 | 6 | topopn 21129 | . . . . . . 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 5044 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
11 | connsubclo.5 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐽) | |
12 | elrestr 16486 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝐵 ∈ 𝐽) → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
13 | 5, 10, 11, 12 | syl3anc 1439 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
14 | connsubclo.6 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅) | |
15 | eqid 2778 | . . . . . 6 ⊢ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴) | |
16 | ineq1 4030 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ∩ 𝐴) = (𝐵 ∩ 𝐴)) | |
17 | 16 | rspceeqv 3529 | . . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴)) → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)) |
18 | 3, 15, 17 | sylancl 580 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)) |
19 | 6 | restcld 21395 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))) |
20 | 5, 9, 19 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))) |
21 | 18, 20 | mpbird 249 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴))) |
22 | 1, 2, 13, 14, 21 | connclo 21638 | . . 3 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = ∪ (𝐽 ↾t 𝐴)) |
23 | 6 | restuni 21385 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
24 | 5, 9, 23 | syl2anc 579 | . . 3 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
25 | 22, 24 | eqtr4d 2817 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴) |
26 | sseqin2 4040 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) | |
27 | 25, 26 | sylibr 226 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃wrex 3091 Vcvv 3398 ∩ cin 3791 ⊆ wss 3792 ∅c0 4141 ∪ cuni 4673 ‘cfv 6137 (class class class)co 6924 ↾t crest 16478 Topctop 21116 Clsdccld 21239 Conncconn 21634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-oadd 7849 df-er 8028 df-en 8244 df-fin 8247 df-fi 8607 df-rest 16480 df-topgen 16501 df-top 21117 df-topon 21134 df-bases 21169 df-cld 21242 df-conn 21635 |
This theorem is referenced by: conncn 21649 conncompclo 21658 |
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