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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 2821 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | cnf 21848 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾) |
7 | 6 | ffund 6512 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
8 | connima.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
9 | 6 | fdmd 6517 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝑋) |
10 | 8, 9 | sseqtrrd 4007 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
11 | fores 6594 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) | |
12 | 7, 10, 11 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
13 | cntop2 21843 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
14 | 2, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top) |
15 | imassrn 5934 | . . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 | |
16 | 6 | frnd 6515 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾) |
17 | 15, 16 | sstrid 3977 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ ∪ 𝐾) |
18 | 4 | restuni 21764 | . . . . 5 ⊢ ((𝐾 ∈ Top ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → (𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
19 | 14, 17, 18 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
20 | foeq3 6582 | . . . 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 234 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
23 | 3 | cnrest 21887 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
24 | 2, 8, 23 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
25 | toptopon2 21520 | . . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
26 | 14, 25 | sylib 220 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
27 | df-ima 5562 | . . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
28 | eqimss2 4023 | . . . . 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 21888 | . . . 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 1367 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴))))) |
32 | 24, 31 | mpbid 234 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴)))) |
33 | eqid 2821 | . . 3 ⊢ ∪ (𝐾 ↾t (𝐹 “ 𝐴)) = ∪ (𝐾 ↾t (𝐹 “ 𝐴)) | |
34 | 33 | cnconn 22024 | . 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 1367 | 1 ⊢ (𝜑 → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Conn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∪ cuni 4831 dom cdm 5549 ran crn 5550 ↾ cres 5551 “ cima 5552 Fun wfun 6343 ⟶wf 6345 –onto→wfo 6347 ‘cfv 6349 (class class class)co 7150 ↾t crest 16688 Topctop 21495 TopOnctopon 21512 Cn ccn 21826 Conncconn 22013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-fin 8507 df-fi 8869 df-rest 16690 df-topgen 16711 df-top 21496 df-topon 21513 df-bases 21548 df-cld 21621 df-cn 21829 df-conn 22014 |
This theorem is referenced by: tgpconncompeqg 22714 tgpconncomp 22715 |
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