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Theorem conncn 22034
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 2801 . . . . 5 𝐾 = 𝐾
42, 3cnf 21854 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
51, 4syl 17 . . 3 (𝜑𝐹:𝑋 𝐾)
65ffnd 6492 . 2 (𝜑𝐹 Fn 𝑋)
75frnd 6498 . . 3 (𝜑 → ran 𝐹 𝐾)
8 conncn.j . . . 4 (𝜑𝐽 ∈ Conn)
9 dffn4 6575 . . . . . 6 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
106, 9sylib 221 . . . . 5 (𝜑𝐹:𝑋onto→ran 𝐹)
11 cntop2 21849 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
121, 11syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
133restuni 21770 . . . . . . 7 ((𝐾 ∈ Top ∧ ran 𝐹 𝐾) → ran 𝐹 = (𝐾t ran 𝐹))
1412, 7, 13syl2anc 587 . . . . . 6 (𝜑 → ran 𝐹 = (𝐾t ran 𝐹))
15 foeq3 6567 . . . . . 6 (ran 𝐹 = (𝐾t ran 𝐹) → (𝐹:𝑋onto→ran 𝐹𝐹:𝑋onto (𝐾t ran 𝐹)))
1614, 15syl 17 . . . . 5 (𝜑 → (𝐹:𝑋onto→ran 𝐹𝐹:𝑋onto (𝐾t ran 𝐹)))
1710, 16mpbid 235 . . . 4 (𝜑𝐹:𝑋onto (𝐾t ran 𝐹))
18 toptopon2 21526 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
1912, 18sylib 221 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
20 ssidd 3941 . . . . . 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 (𝐾t ran 𝐹))))
2219, 20, 7, 21syl3anc 1368 . . . . 5 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))))
231, 22mpbid 235 . . . 4 (𝜑𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹)))
24 eqid 2801 . . . . 5 (𝐾t ran 𝐹) = (𝐾t ran 𝐹)
2524cnconn 22030 . . . 4 ((𝐽 ∈ Conn ∧ 𝐹:𝑋onto (𝐾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))) → (𝐾t ran 𝐹) ∈ Conn)
268, 17, 23, 25syl3anc 1368 . . 3 (𝜑 → (𝐾t ran 𝐹) ∈ Conn)
27 conncn.u . . 3 (𝜑𝑈𝐾)
28 conncn.1 . . . 4 (𝜑 → (𝐹𝐴) ∈ 𝑈)
29 conncn.a . . . . 5 (𝜑𝐴𝑋)
30 fnfvelrn 6829 . . . . 5 ((𝐹 Fn 𝑋𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
316, 29, 30syl2anc 587 . . . 4 (𝜑 → (𝐹𝐴) ∈ ran 𝐹)
32 inelcm 4375 . . . 4 (((𝐹𝐴) ∈ 𝑈 ∧ (𝐹𝐴) ∈ ran 𝐹) → (𝑈 ∩ ran 𝐹) ≠ ∅)
3328, 31, 32syl2anc 587 . . 3 (𝜑 → (𝑈 ∩ ran 𝐹) ≠ ∅)
34 conncn.c . . 3 (𝜑𝑈 ∈ (Clsd‘𝐾))
353, 7, 26, 27, 33, 34connsubclo 22032 . 2 (𝜑 → ran 𝐹𝑈)
36 df-f 6332 . 2 (𝐹:𝑋𝑈 ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹𝑈))
376, 35, 36sylanbrc 586 1 (𝜑𝐹:𝑋𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2112  wne 2990  cin 3883  wss 3884  c0 4246   cuni 4803  ran crn 5524   Fn wfn 6323  wf 6324  ontowfo 6326  cfv 6328  (class class class)co 7139  t crest 16689  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 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-fin 8500  df-fi 8863  df-rest 16691  df-topgen 16712  df-top 21502  df-topon 21519  df-bases 21554  df-cld 21627  df-cn 21835  df-conn 22020
This theorem is referenced by:  pconnconn  32586  cvmliftmolem1  32636  cvmlift2lem9  32666  cvmlift3lem6  32679
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