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Mirrors > Home > MPE Home > Th. List > conncn | Structured version Visualization version GIF version |
Description: A continuous function from a connected topology with one point in a clopen set must lie entirely within the set. (Contributed by Mario Carneiro, 16-Feb-2015.) |
Ref | Expression |
---|---|
conncn.x | ⊢ 𝑋 = ∪ 𝐽 |
conncn.j | ⊢ (𝜑 → 𝐽 ∈ Conn) |
conncn.f | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
conncn.u | ⊢ (𝜑 → 𝑈 ∈ 𝐾) |
conncn.c | ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐾)) |
conncn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
conncn.1 | ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑈) |
Ref | Expression |
---|---|
conncn | ⊢ (𝜑 → 𝐹:𝑋⟶𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | conncn.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
2 | conncn.x | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | eqid 2736 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
4 | 2, 3 | cnf 22097 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾) |
6 | 5 | ffnd 6524 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝑋) |
7 | 5 | frnd 6531 | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾) |
8 | conncn.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Conn) | |
9 | dffn4 6617 | . . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹) | |
10 | 6, 9 | sylib 221 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹) |
11 | cntop2 22092 | . . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
12 | 1, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top) |
13 | 3 | restuni 22013 | . . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ ran 𝐹 ⊆ ∪ 𝐾) → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹)) |
14 | 12, 7, 13 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹)) |
15 | foeq3 6609 | . . . . . 6 ⊢ (ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹) → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹))) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹))) |
17 | 10, 16 | mpbid 235 | . . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹)) |
18 | toptopon2 21769 | . . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
19 | 12, 18 | sylib 221 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
20 | ssidd 3910 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ran 𝐹) | |
21 | cnrest2 22137 | . . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ ∪ 𝐾) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))) | |
22 | 19, 20, 7, 21 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))) |
23 | 1, 22 | mpbid 235 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))) |
24 | eqid 2736 | . . . . 5 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹) | |
25 | 24 | cnconn 22273 | . . . 4 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))) → (𝐾 ↾t ran 𝐹) ∈ Conn) |
26 | 8, 17, 23, 25 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝐾 ↾t ran 𝐹) ∈ Conn) |
27 | conncn.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐾) | |
28 | conncn.1 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑈) | |
29 | conncn.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
30 | fnfvelrn 6879 | . . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹) | |
31 | 6, 29, 30 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹) |
32 | inelcm 4365 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ 𝑈 ∧ (𝐹‘𝐴) ∈ ran 𝐹) → (𝑈 ∩ ran 𝐹) ≠ ∅) | |
33 | 28, 31, 32 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑈 ∩ ran 𝐹) ≠ ∅) |
34 | conncn.c | . . 3 ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐾)) | |
35 | 3, 7, 26, 27, 33, 34 | connsubclo 22275 | . 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑈) |
36 | df-f 6362 | . 2 ⊢ (𝐹:𝑋⟶𝑈 ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹 ⊆ 𝑈)) | |
37 | 6, 35, 36 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐹:𝑋⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 ∪ cuni 4805 ran crn 5537 Fn wfn 6353 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 (class class class)co 7191 ↾t crest 16879 Topctop 21744 TopOnctopon 21761 Clsdccld 21867 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: pconnconn 32860 cvmliftmolem1 32910 cvmlift2lem9 32940 cvmlift3lem6 32953 |
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