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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 2736 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | cnf 22097 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾) |
7 | 6 | ffund 6527 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
8 | connima.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
9 | 6 | fdmd 6534 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝑋) |
10 | 8, 9 | sseqtrrd 3928 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
11 | fores 6621 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) | |
12 | 7, 10, 11 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
13 | cntop2 22092 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
14 | 2, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top) |
15 | imassrn 5925 | . . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 | |
16 | 6 | frnd 6531 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾) |
17 | 15, 16 | sstrid 3898 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ ∪ 𝐾) |
18 | 4 | restuni 22013 | . . . . 5 ⊢ ((𝐾 ∈ Top ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → (𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
19 | 14, 17, 18 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
20 | foeq3 6609 | . . . 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 235 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴))) |
23 | 3 | cnrest 22136 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
24 | 2, 8, 23 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
25 | toptopon2 21769 | . . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
26 | 14, 25 | sylib 221 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
27 | df-ima 5549 | . . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
28 | eqimss2 3944 | . . . . 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 22137 | . . . 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 1373 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴))))) |
32 | 24, 31 | mpbid 235 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴)))) |
33 | eqid 2736 | . . 3 ⊢ ∪ (𝐾 ↾t (𝐹 “ 𝐴)) = ∪ (𝐾 ↾t (𝐹 “ 𝐴)) | |
34 | 33 | cnconn 22273 | . 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 1373 | 1 ⊢ (𝜑 → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Conn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 ∪ cuni 4805 dom cdm 5536 ran crn 5537 ↾ cres 5538 “ cima 5539 Fun wfun 6352 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 (class class class)co 7191 ↾t crest 16879 Topctop 21744 TopOnctopon 21761 Cn ccn 22075 Conncconn 22262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-map 8488 df-en 8605 df-fin 8608 df-fi 9005 df-rest 16881 df-topgen 16902 df-top 21745 df-topon 21762 df-bases 21797 df-cld 21870 df-cn 22078 df-conn 22263 |
This theorem is referenced by: tgpconncompeqg 22963 tgpconncomp 22964 |
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