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Theorem concn 21039
 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
concn.x 𝑋 = 𝐽
concn.j (𝜑𝐽 ∈ Con)
concn.f (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
concn.u (𝜑𝑈𝐾)
concn.c (𝜑𝑈 ∈ (Clsd‘𝐾))
concn.a (𝜑𝐴𝑋)
concn.1 (𝜑 → (𝐹𝐴) ∈ 𝑈)
Assertion
Ref Expression
concn (𝜑𝐹:𝑋𝑈)

Proof of Theorem concn
StepHypRef Expression
1 concn.f . . . 4 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
2 concn.x . . . . 5 𝑋 = 𝐽
3 eqid 2610 . . . . 5 𝐾 = 𝐾
42, 3cnf 20860 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
51, 4syl 17 . . 3 (𝜑𝐹:𝑋 𝐾)
6 ffn 5958 . . 3 (𝐹:𝑋 𝐾𝐹 Fn 𝑋)
75, 6syl 17 . 2 (𝜑𝐹 Fn 𝑋)
8 frn 5966 . . . 4 (𝐹:𝑋 𝐾 → ran 𝐹 𝐾)
95, 8syl 17 . . 3 (𝜑 → ran 𝐹 𝐾)
10 concn.j . . . 4 (𝜑𝐽 ∈ Con)
11 dffn4 6034 . . . . . 6 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
127, 11sylib 207 . . . . 5 (𝜑𝐹:𝑋onto→ran 𝐹)
13 cntop2 20855 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
141, 13syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
153restuni 20776 . . . . . . 7 ((𝐾 ∈ Top ∧ ran 𝐹 𝐾) → ran 𝐹 = (𝐾t ran 𝐹))
1614, 9, 15syl2anc 691 . . . . . 6 (𝜑 → ran 𝐹 = (𝐾t ran 𝐹))
17 foeq3 6026 . . . . . 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 221 . . . 4 (𝜑𝐹:𝑋onto (𝐾t ran 𝐹))
203toptopon 20548 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
2114, 20sylib 207 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
22 ssid 3587 . . . . . . 7 ran 𝐹 ⊆ ran 𝐹
2322a1i 11 . . . . . 6 (𝜑 → ran 𝐹 ⊆ ran 𝐹)
24 cnrest2 20900 . . . . . 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 1318 . . . . 5 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))))
261, 25mpbid 221 . . . 4 (𝜑𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹)))
27 eqid 2610 . . . . 5 (𝐾t ran 𝐹) = (𝐾t ran 𝐹)
2827cnconn 21035 . . . 4 ((𝐽 ∈ Con ∧ 𝐹:𝑋onto (𝐾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))) → (𝐾t ran 𝐹) ∈ Con)
2910, 19, 26, 28syl3anc 1318 . . 3 (𝜑 → (𝐾t ran 𝐹) ∈ Con)
30 concn.u . . 3 (𝜑𝑈𝐾)
31 concn.1 . . . 4 (𝜑 → (𝐹𝐴) ∈ 𝑈)
32 concn.a . . . . 5 (𝜑𝐴𝑋)
33 fnfvelrn 6264 . . . . 5 ((𝐹 Fn 𝑋𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
347, 32, 33syl2anc 691 . . . 4 (𝜑 → (𝐹𝐴) ∈ ran 𝐹)
35 inelcm 3984 . . . 4 (((𝐹𝐴) ∈ 𝑈 ∧ (𝐹𝐴) ∈ ran 𝐹) → (𝑈 ∩ ran 𝐹) ≠ ∅)
3631, 34, 35syl2anc 691 . . 3 (𝜑 → (𝑈 ∩ ran 𝐹) ≠ ∅)
37 concn.c . . 3 (𝜑𝑈 ∈ (Clsd‘𝐾))
383, 9, 29, 30, 36, 37consubclo 21037 . 2 (𝜑 → ran 𝐹𝑈)
39 df-f 5808 . 2 (𝐹:𝑋𝑈 ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹𝑈))
407, 38, 39sylanbrc 695 1 (𝜑𝐹:𝑋𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  TopOnctopon 20518  Clsdccld 20630   Cn ccn 20838  Conccon 21024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-cn 20841  df-con 21025 This theorem is referenced by:  pconcon  30467  cvmliftmolem1  30517  cvmlift2lem9  30547  cvmlift3lem6  30560
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