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Theorem conncn 21277
 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.)
Hypotheses
Ref Expression
conncn.x 𝑋 = 𝐽
conncn.j (𝜑𝐽 ∈ Conn)
conncn.f (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
conncn.u (𝜑𝑈𝐾)
conncn.c (𝜑𝑈 ∈ (Clsd‘𝐾))
conncn.a (𝜑𝐴𝑋)
conncn.1 (𝜑 → (𝐹𝐴) ∈ 𝑈)
Assertion
Ref Expression
conncn (𝜑𝐹:𝑋𝑈)

Proof of Theorem conncn
StepHypRef Expression
1 conncn.f . . . 4 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
2 conncn.x . . . . 5 𝑋 = 𝐽
3 eqid 2651 . . . . 5 𝐾 = 𝐾
42, 3cnf 21098 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
51, 4syl 17 . . 3 (𝜑𝐹:𝑋 𝐾)
6 ffn 6083 . . 3 (𝐹:𝑋 𝐾𝐹 Fn 𝑋)
75, 6syl 17 . 2 (𝜑𝐹 Fn 𝑋)
8 frn 6091 . . . 4 (𝐹:𝑋 𝐾 → ran 𝐹 𝐾)
95, 8syl 17 . . 3 (𝜑 → ran 𝐹 𝐾)
10 conncn.j . . . 4 (𝜑𝐽 ∈ Conn)
11 dffn4 6159 . . . . . 6 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
127, 11sylib 208 . . . . 5 (𝜑𝐹:𝑋onto→ran 𝐹)
13 cntop2 21093 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
141, 13syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
153restuni 21014 . . . . . . 7 ((𝐾 ∈ Top ∧ ran 𝐹 𝐾) → ran 𝐹 = (𝐾t ran 𝐹))
1614, 9, 15syl2anc 694 . . . . . 6 (𝜑 → ran 𝐹 = (𝐾t ran 𝐹))
17 foeq3 6151 . . . . . 6 (ran 𝐹 = (𝐾t ran 𝐹) → (𝐹:𝑋onto→ran 𝐹𝐹:𝑋onto (𝐾t ran 𝐹)))
1816, 17syl 17 . . . . 5 (𝜑 → (𝐹:𝑋onto→ran 𝐹𝐹:𝑋onto (𝐾t ran 𝐹)))
1912, 18mpbid 222 . . . 4 (𝜑𝐹:𝑋onto (𝐾t ran 𝐹))
203toptopon 20770 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
2114, 20sylib 208 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
22 ssid 3657 . . . . . . 7 ran 𝐹 ⊆ ran 𝐹
2322a1i 11 . . . . . 6 (𝜑 → ran 𝐹 ⊆ ran 𝐹)
24 cnrest2 21138 . . . . . 6 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 𝐾) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))))
2521, 23, 9, 24syl3anc 1366 . . . . 5 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))))
261, 25mpbid 222 . . . 4 (𝜑𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹)))
27 eqid 2651 . . . . 5 (𝐾t ran 𝐹) = (𝐾t ran 𝐹)
2827cnconn 21273 . . . 4 ((𝐽 ∈ Conn ∧ 𝐹:𝑋onto (𝐾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))) → (𝐾t ran 𝐹) ∈ Conn)
2910, 19, 26, 28syl3anc 1366 . . 3 (𝜑 → (𝐾t ran 𝐹) ∈ Conn)
30 conncn.u . . 3 (𝜑𝑈𝐾)
31 conncn.1 . . . 4 (𝜑 → (𝐹𝐴) ∈ 𝑈)
32 conncn.a . . . . 5 (𝜑𝐴𝑋)
33 fnfvelrn 6396 . . . . 5 ((𝐹 Fn 𝑋𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
347, 32, 33syl2anc 694 . . . 4 (𝜑 → (𝐹𝐴) ∈ ran 𝐹)
35 inelcm 4065 . . . 4 (((𝐹𝐴) ∈ 𝑈 ∧ (𝐹𝐴) ∈ ran 𝐹) → (𝑈 ∩ ran 𝐹) ≠ ∅)
3631, 34, 35syl2anc 694 . . 3 (𝜑 → (𝑈 ∩ ran 𝐹) ≠ ∅)
37 conncn.c . . 3 (𝜑𝑈 ∈ (Clsd‘𝐾))
383, 9, 29, 30, 36, 37connsubclo 21275 . 2 (𝜑 → ran 𝐹𝑈)
39 df-f 5930 . 2 (𝐹:𝑋𝑈 ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹𝑈))
407, 38, 39sylanbrc 699 1 (𝜑𝐹:𝑋𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1523   ∈ wcel 2030   ≠ wne 2823   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  ∪ cuni 4468  ran crn 5144   Fn wfn 5921  ⟶wf 5922  –onto→wfo 5924  ‘cfv 5926  (class class class)co 6690   ↾t crest 16128  Topctop 20746  TopOnctopon 20763  Clsdccld 20868   Cn ccn 21076  Conncconn 21262 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-cn 21079  df-conn 21263 This theorem is referenced by:  pconnconn  31339  cvmliftmolem1  31389  cvmlift2lem9  31419  cvmlift3lem6  31432
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