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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 2821 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
4 | 2, 3 | cnf 21854 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾) |
6 | 5 | ffnd 6515 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝑋) |
7 | 5 | frnd 6521 | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾) |
8 | conncn.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Conn) | |
9 | dffn4 6596 | . . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹) | |
10 | 6, 9 | sylib 220 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹) |
11 | cntop2 21849 | . . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
12 | 1, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top) |
13 | 3 | restuni 21770 | . . . . . . 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 6588 | . . . . . 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 21526 | . . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
19 | 12, 18 | sylib 220 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
20 | ssidd 3990 | . . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ran 𝐹) | |
21 | cnrest2 21894 | . . . . . 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 2821 | . . . . 5 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹) | |
25 | 24 | cnconn 22030 | . . . 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 6848 | . . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹) | |
31 | 6, 29, 30 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹) |
32 | inelcm 4414 | . . . 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 22032 | . 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑈) |
36 | df-f 6359 | . 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 3016 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4838 ran crn 5556 Fn wfn 6350 ⟶wf 6351 –onto→wfo 6353 ‘cfv 6355 (class class class)co 7156 ↾t crest 16694 Topctop 21501 TopOnctopon 21518 Clsdccld 21624 Cn ccn 21832 Conncconn 22019 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-fin 8513 df-fi 8875 df-rest 16696 df-topgen 16717 df-top 21502 df-topon 21519 df-bases 21554 df-cld 21627 df-cn 21835 df-conn 22020 |
This theorem is referenced by: pconnconn 32478 cvmliftmolem1 32528 cvmlift2lem9 32558 cvmlift3lem6 32571 |
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