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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 2726 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | cnf 23235 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾) |
7 | 6 | ffund 6721 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
8 | connima.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
9 | 6 | fdmd 6727 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝑋) |
10 | 8, 9 | sseqtrrd 4020 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
11 | fores 6814 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) | |
12 | 7, 10, 11 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
13 | cntop2 23230 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
14 | 2, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top) |
15 | imassrn 6070 | . . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 | |
16 | 6 | frnd 6725 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾) |
17 | 15, 16 | sstrid 3990 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ ∪ 𝐾) |
18 | 4 | restuni 23151 | . . . . 5 ⊢ ((𝐾 ∈ Top ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → (𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
19 | 14, 17, 18 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
20 | foeq3 6802 | . . . 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 231 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
23 | 3 | cnrest 23274 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
24 | 2, 8, 23 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
25 | toptopon2 22905 | . . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
26 | 14, 25 | sylib 217 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
27 | df-ima 5685 | . . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
28 | eqimss2 4038 | . . . . 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 23275 | . . . 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 1368 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴))))) |
32 | 24, 31 | mpbid 231 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴)))) |
33 | eqid 2726 | . . 3 ⊢ ∪ (𝐾 ↾t (𝐹 “ 𝐴)) = ∪ (𝐾 ↾t (𝐹 “ 𝐴)) | |
34 | 33 | cnconn 23411 | . 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 1368 | 1 ⊢ (𝜑 → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Conn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ⊆ wss 3946 ∪ cuni 4905 dom cdm 5672 ran crn 5673 ↾ cres 5674 “ cima 5675 Fun wfun 6537 ⟶wf 6539 –onto→wfo 6541 ‘cfv 6543 (class class class)co 7413 ↾t crest 17427 Topctop 22880 TopOnctopon 22897 Cn ccn 23213 Conncconn 23400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-map 8846 df-en 8964 df-fin 8967 df-fi 9444 df-rest 17429 df-topgen 17450 df-top 22881 df-topon 22898 df-bases 22934 df-cld 23008 df-cn 23216 df-conn 23401 |
This theorem is referenced by: tgpconncompeqg 24101 tgpconncomp 24102 |
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