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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 2622 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
4 | 2, 3 | cnf 21050 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾) |
6 | ffn 6045 | . . 3 ⊢ (𝐹:𝑋⟶∪ 𝐾 → 𝐹 Fn 𝑋) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝑋) |
8 | frn 6053 | . . . 4 ⊢ (𝐹:𝑋⟶∪ 𝐾 → ran 𝐹 ⊆ ∪ 𝐾) | |
9 | 5, 8 | syl 17 | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾) |
10 | conncn.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Conn) | |
11 | dffn4 6121 | . . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹) | |
12 | 7, 11 | sylib 208 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹) |
13 | cntop2 21045 | . . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
14 | 1, 13 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top) |
15 | 3 | restuni 20966 | . . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ ran 𝐹 ⊆ ∪ 𝐾) → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹)) |
16 | 14, 9, 15 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹)) |
17 | foeq3 6113 | . . . . . 6 ⊢ (ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹) → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹))) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹))) |
19 | 12, 18 | mpbid 222 | . . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹)) |
20 | 3 | toptopon 20722 | . . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
21 | 14, 20 | sylib 208 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
22 | ssid 3624 | . . . . . . 7 ⊢ ran 𝐹 ⊆ ran 𝐹 | |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ran 𝐹) |
24 | cnrest2 21090 | . . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ ∪ 𝐾) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))) | |
25 | 21, 23, 9, 24 | syl3anc 1326 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))) |
26 | 1, 25 | mpbid 222 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))) |
27 | eqid 2622 | . . . . 5 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹) | |
28 | 27 | cnconn 21225 | . . . 4 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))) → (𝐾 ↾t ran 𝐹) ∈ Conn) |
29 | 10, 19, 26, 28 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (𝐾 ↾t ran 𝐹) ∈ Conn) |
30 | conncn.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐾) | |
31 | conncn.1 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑈) | |
32 | conncn.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
33 | fnfvelrn 6356 | . . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹) | |
34 | 7, 32, 33 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹) |
35 | inelcm 4032 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ 𝑈 ∧ (𝐹‘𝐴) ∈ ran 𝐹) → (𝑈 ∩ ran 𝐹) ≠ ∅) | |
36 | 31, 34, 35 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝑈 ∩ ran 𝐹) ≠ ∅) |
37 | conncn.c | . . 3 ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐾)) | |
38 | 3, 9, 29, 30, 36, 37 | connsubclo 21227 | . 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑈) |
39 | df-f 5892 | . 2 ⊢ (𝐹:𝑋⟶𝑈 ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹 ⊆ 𝑈)) | |
40 | 7, 38, 39 | sylanbrc 698 | 1 ⊢ (𝜑 → 𝐹:𝑋⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 ∪ cuni 4436 ran crn 5115 Fn wfn 5883 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 ↾t crest 16081 Topctop 20698 TopOnctopon 20715 Clsdccld 20820 Cn ccn 21028 Conncconn 21214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-fin 7959 df-fi 8317 df-rest 16083 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-cn 21031 df-conn 21215 |
This theorem is referenced by: pconnconn 31213 cvmliftmolem1 31263 cvmlift2lem9 31293 cvmlift3lem6 31306 |
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