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Theorem conncn 21210
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 2620 . . . . 5 𝐾 = 𝐾
42, 3cnf 21031 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
51, 4syl 17 . . 3 (𝜑𝐹:𝑋 𝐾)
6 ffn 6032 . . 3 (𝐹:𝑋 𝐾𝐹 Fn 𝑋)
75, 6syl 17 . 2 (𝜑𝐹 Fn 𝑋)
8 frn 6040 . . . 4 (𝐹:𝑋 𝐾 → ran 𝐹 𝐾)
95, 8syl 17 . . 3 (𝜑 → ran 𝐹 𝐾)
10 conncn.j . . . 4 (𝜑𝐽 ∈ Conn)
11 dffn4 6108 . . . . . 6 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
127, 11sylib 208 . . . . 5 (𝜑𝐹:𝑋onto→ran 𝐹)
13 cntop2 21026 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
141, 13syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
153restuni 20947 . . . . . . 7 ((𝐾 ∈ Top ∧ ran 𝐹 𝐾) → ran 𝐹 = (𝐾t ran 𝐹))
1614, 9, 15syl2anc 692 . . . . . 6 (𝜑 → ran 𝐹 = (𝐾t ran 𝐹))
17 foeq3 6100 . . . . . 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 20703 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
2114, 20sylib 208 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
22 ssid 3616 . . . . . . 7 ran 𝐹 ⊆ ran 𝐹
2322a1i 11 . . . . . 6 (𝜑 → ran 𝐹 ⊆ ran 𝐹)
24 cnrest2 21071 . . . . . 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 1324 . . . . 5 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))))
261, 25mpbid 222 . . . 4 (𝜑𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹)))
27 eqid 2620 . . . . 5 (𝐾t ran 𝐹) = (𝐾t ran 𝐹)
2827cnconn 21206 . . . 4 ((𝐽 ∈ Conn ∧ 𝐹:𝑋onto (𝐾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))) → (𝐾t ran 𝐹) ∈ Conn)
2910, 19, 26, 28syl3anc 1324 . . 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 6342 . . . . 5 ((𝐹 Fn 𝑋𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
347, 32, 33syl2anc 692 . . . 4 (𝜑 → (𝐹𝐴) ∈ ran 𝐹)
35 inelcm 4023 . . . 4 (((𝐹𝐴) ∈ 𝑈 ∧ (𝐹𝐴) ∈ ran 𝐹) → (𝑈 ∩ ran 𝐹) ≠ ∅)
3631, 34, 35syl2anc 692 . . 3 (𝜑 → (𝑈 ∩ ran 𝐹) ≠ ∅)
37 conncn.c . . 3 (𝜑𝑈 ∈ (Clsd‘𝐾))
383, 9, 29, 30, 36, 37connsubclo 21208 . 2 (𝜑 → ran 𝐹𝑈)
39 df-f 5880 . 2 (𝐹:𝑋𝑈 ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹𝑈))
407, 38, 39sylanbrc 697 1 (𝜑𝐹:𝑋𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1481  wcel 1988  wne 2791  cin 3566  wss 3567  c0 3907   cuni 4427  ran crn 5105   Fn wfn 5871  wf 5872  ontowfo 5874  cfv 5876  (class class class)co 6635  t crest 16062  Topctop 20679  TopOnctopon 20696  Clsdccld 20801   Cn ccn 21009  Conncconn 21195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-fin 7944  df-fi 8302  df-rest 16064  df-topgen 16085  df-top 20680  df-topon 20697  df-bases 20731  df-cld 20804  df-cn 21012  df-conn 21196
This theorem is referenced by:  pconnconn  31187  cvmliftmolem1  31237  cvmlift2lem9  31267  cvmlift3lem6  31280
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