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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 2823 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | cnf 21856 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾) |
7 | 6 | ffund 6520 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
8 | connima.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
9 | 6 | fdmd 6525 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝑋) |
10 | 8, 9 | sseqtrrd 4010 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
11 | fores 6602 | . . . 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 21851 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
14 | 2, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top) |
15 | imassrn 5942 | . . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 | |
16 | 6 | frnd 6523 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾) |
17 | 15, 16 | sstrid 3980 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ ∪ 𝐾) |
18 | 4 | restuni 21772 | . . . . 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 6590 | . . . 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 21895 | . . . 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 21528 | . . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
26 | 14, 25 | sylib 220 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
27 | df-ima 5570 | . . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
28 | eqimss2 4026 | . . . . 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 21896 | . . . 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 2823 | . . 3 ⊢ ∪ (𝐾 ↾t (𝐹 “ 𝐴)) = ∪ (𝐾 ↾t (𝐹 “ 𝐴)) | |
34 | 33 | cnconn 22032 | . 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 1537 ∈ wcel 2114 ⊆ wss 3938 ∪ cuni 4840 dom cdm 5557 ran crn 5558 ↾ cres 5559 “ cima 5560 Fun wfun 6351 ⟶wf 6353 –onto→wfo 6355 ‘cfv 6357 (class class class)co 7158 ↾t crest 16696 Topctop 21503 TopOnctopon 21520 Cn ccn 21834 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-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-map 8410 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-cn 21837 df-conn 22022 |
This theorem is referenced by: tgpconncompeqg 22722 tgpconncomp 22723 |
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