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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 2820 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
4 | 2, 3 | cnf 21832 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾) |
6 | 5 | ffnd 6496 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝑋) |
7 | 5 | frnd 6502 | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾) |
8 | conncn.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Conn) | |
9 | dffn4 6577 | . . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹) | |
10 | 6, 9 | sylib 220 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹) |
11 | cntop2 21827 | . . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
12 | 1, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top) |
13 | 3 | restuni 21748 | . . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ ran 𝐹 ⊆ ∪ 𝐾) → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹)) |
14 | 12, 7, 13 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹)) |
15 | foeq3 6569 | . . . . . 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 234 | . . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹)) |
18 | toptopon2 21504 | . . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
19 | 12, 18 | sylib 220 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
20 | ssidd 3973 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ran 𝐹) | |
21 | cnrest2 21872 | . . . . . 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 1367 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))) |
23 | 1, 22 | mpbid 234 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))) |
24 | eqid 2820 | . . . . 5 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹) | |
25 | 24 | cnconn 22008 | . . . 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 1367 | . . 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 6829 | . . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹) | |
31 | 6, 29, 30 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹) |
32 | inelcm 4395 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ 𝑈 ∧ (𝐹‘𝐴) ∈ ran 𝐹) → (𝑈 ∩ ran 𝐹) ≠ ∅) | |
33 | 28, 31, 32 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑈 ∩ ran 𝐹) ≠ ∅) |
34 | conncn.c | . . 3 ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐾)) | |
35 | 3, 7, 26, 27, 33, 34 | connsubclo 22010 | . 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑈) |
36 | df-f 6340 | . 2 ⊢ (𝐹:𝑋⟶𝑈 ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹 ⊆ 𝑈)) | |
37 | 6, 35, 36 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐹:𝑋⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3011 ∩ cin 3918 ⊆ wss 3919 ∅c0 4274 ∪ cuni 4819 ran crn 5537 Fn wfn 6331 ⟶wf 6332 –onto→wfo 6334 ‘cfv 6336 (class class class)co 7137 ↾t crest 16672 Topctop 21479 TopOnctopon 21496 Clsdccld 21602 Cn ccn 21810 Conncconn 21997 |
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 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 |
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 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 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-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-oadd 8087 df-er 8270 df-map 8389 df-en 8491 df-fin 8494 df-fi 8856 df-rest 16674 df-topgen 16695 df-top 21480 df-topon 21497 df-bases 21532 df-cld 21605 df-cn 21813 df-conn 21998 |
This theorem is referenced by: pconnconn 32480 cvmliftmolem1 32530 cvmlift2lem9 32560 cvmlift3lem6 32573 |
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