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Mirrors > Home > MPE Home > Th. List > connima | Structured version Visualization version GIF version |
Description: The image of a connected set is connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
connima.x | ⊢ 𝑋 = ∪ 𝐽 |
connima.f | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
connima.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
connima.c | ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ Conn) |
Ref | Expression |
---|---|
connima | ⊢ (𝜑 → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Conn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | connima.c | . 2 ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ Conn) | |
2 | connima.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
3 | connima.x | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
4 | eqid 2777 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | cnf 21458 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾) |
7 | 6 | ffund 6295 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
8 | connima.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
9 | 6 | fdmd 6300 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝑋) |
10 | 8, 9 | sseqtr4d 3860 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
11 | fores 6376 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) | |
12 | 7, 10, 11 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
13 | cntop2 21453 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
14 | 2, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top) |
15 | imassrn 5731 | . . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 | |
16 | 6 | frnd 6298 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾) |
17 | 15, 16 | syl5ss 3831 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ ∪ 𝐾) |
18 | 4 | restuni 21374 | . . . . 5 ⊢ ((𝐾 ∈ Top ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → (𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
19 | 14, 17, 18 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
20 | foeq3 6364 | . . . 4 ⊢ ((𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴)) → ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴)))) | |
21 | 19, 20 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴)))) |
22 | 12, 21 | mpbid 224 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
23 | 3 | cnrest 21497 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
24 | 2, 8, 23 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
25 | 4 | toptopon 21129 | . . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
26 | 14, 25 | sylib 210 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
27 | df-ima 5368 | . . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
28 | eqimss2 3876 | . . . . 5 ⊢ ((𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) → ran (𝐹 ↾ 𝐴) ⊆ (𝐹 “ 𝐴)) | |
29 | 27, 28 | mp1i 13 | . . . 4 ⊢ (𝜑 → ran (𝐹 ↾ 𝐴) ⊆ (𝐹 “ 𝐴)) |
30 | cnrest2 21498 | . . . 4 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ran (𝐹 ↾ 𝐴) ⊆ (𝐹 “ 𝐴) ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴))))) | |
31 | 26, 29, 17, 30 | syl3anc 1439 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴))))) |
32 | 24, 31 | mpbid 224 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴)))) |
33 | eqid 2777 | . . 3 ⊢ ∪ (𝐾 ↾t (𝐹 “ 𝐴)) = ∪ (𝐾 ↾t (𝐹 “ 𝐴)) | |
34 | 33 | cnconn 21634 | . 2 ⊢ (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴)) ∧ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴)))) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Conn) |
35 | 1, 22, 32, 34 | syl3anc 1439 | 1 ⊢ (𝜑 → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Conn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 ∪ cuni 4671 dom cdm 5355 ran crn 5356 ↾ cres 5357 “ cima 5358 Fun wfun 6129 ⟶wf 6131 –onto→wfo 6133 ‘cfv 6135 (class class class)co 6922 ↾t crest 16467 Topctop 21105 TopOnctopon 21122 Cn ccn 21436 Conncconn 21623 |
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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-fin 8245 df-fi 8605 df-rest 16469 df-topgen 16490 df-top 21106 df-topon 21123 df-bases 21158 df-cld 21231 df-cn 21439 df-conn 21624 |
This theorem is referenced by: tgpconncompeqg 22323 tgpconncomp 22324 |
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