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Mirrors > Home > MPE Home > Th. List > conncompclo | Structured version Visualization version GIF version |
Description: The connected component containing 𝐴 is a subset of any clopen set containing 𝐴. (Contributed by Mario Carneiro, 20-Sep-2015.) |
Ref | Expression |
---|---|
conncomp.2 | ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} |
Ref | Expression |
---|---|
conncompclo | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ⊆ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | simp1 1132 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐽 ∈ (TopOn‘𝑋)) | |
3 | simp2 1133 | . . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽))) | |
4 | 3 | elin1d 4177 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ 𝐽) |
5 | toponss 21537 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ 𝐽) → 𝑇 ⊆ 𝑋) | |
6 | 2, 4, 5 | syl2anc 586 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ 𝑋) |
7 | simp3 1134 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇) | |
8 | 6, 7 | sseldd 3970 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑋) |
9 | conncomp.2 | . . . . 5 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} | |
10 | 9 | conncompcld 22044 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝑆 ∈ (Clsd‘𝐽)) |
11 | 2, 8, 10 | syl2anc 586 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ∈ (Clsd‘𝐽)) |
12 | 1 | cldss 21639 | . . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽) |
13 | 11, 12 | syl 17 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ⊆ ∪ 𝐽) |
14 | 9 | conncompconn 22042 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t 𝑆) ∈ Conn) |
15 | 2, 8, 14 | syl2anc 586 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → (𝐽 ↾t 𝑆) ∈ Conn) |
16 | 9 | conncompid 22041 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑆) |
17 | 2, 8, 16 | syl2anc 586 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑆) |
18 | inelcm 4416 | . . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑆) → (𝑇 ∩ 𝑆) ≠ ∅) | |
19 | 7, 17, 18 | syl2anc 586 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → (𝑇 ∩ 𝑆) ≠ ∅) |
20 | 3 | elin2d 4178 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ (Clsd‘𝐽)) |
21 | 1, 13, 15, 4, 19, 20 | connsubclo 22034 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {crab 3144 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 ∪ cuni 4840 ‘cfv 6357 (class class class)co 7158 ↾t crest 16696 TopOnctopon 21520 Clsdccld 21626 Conncconn 22021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-er 8291 df-en 8512 df-fin 8515 df-fi 8877 df-rest 16698 df-topgen 16719 df-top 21504 df-topon 21521 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-conn 22022 |
This theorem is referenced by: tgpconncompss 22724 |
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