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Theorem conncn 23274
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 = βˆͺ 𝐽
conncn.j (πœ‘ β†’ 𝐽 ∈ Conn)
conncn.f (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
conncn.u (πœ‘ β†’ π‘ˆ ∈ 𝐾)
conncn.c (πœ‘ β†’ π‘ˆ ∈ (Clsdβ€˜πΎ))
conncn.a (πœ‘ β†’ 𝐴 ∈ 𝑋)
conncn.1 (πœ‘ β†’ (πΉβ€˜π΄) ∈ π‘ˆ)
Assertion
Ref Expression
conncn (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘ˆ)
Proof of Theorem conncn
StepHypRef Expression
1 conncn.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
2 conncn.x . . . . 5 𝑋 = βˆͺ 𝐽
3 eqid 2724 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
42, 3cnf 23094 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
51, 4syl 17 . . 3 (πœ‘ β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
65ffnd 6709 . 2 (πœ‘ β†’ 𝐹 Fn 𝑋)
75frnd 6716 . . 3 (πœ‘ β†’ ran 𝐹 βŠ† βˆͺ 𝐾)
8 conncn.j . . . 4 (πœ‘ β†’ 𝐽 ∈ Conn)
9 dffn4 6802 . . . . . 6 (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–ontoβ†’ran 𝐹)
106, 9sylib 217 . . . . 5 (πœ‘ β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
11 cntop2 23089 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
121, 11syl 17 . . . . . . 7 (πœ‘ β†’ 𝐾 ∈ Top)
133restuni 23010 . . . . . . 7 ((𝐾 ∈ Top ∧ ran 𝐹 βŠ† βˆͺ 𝐾) β†’ ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹))
1412, 7, 13syl2anc 583 . . . . . 6 (πœ‘ β†’ ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹))
15 foeq3 6794 . . . . . 6 (ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹) β†’ (𝐹:𝑋–ontoβ†’ran 𝐹 ↔ 𝐹:𝑋–ontoβ†’βˆͺ (𝐾 β†Ύt ran 𝐹)))
1614, 15syl 17 . . . . 5 (πœ‘ β†’ (𝐹:𝑋–ontoβ†’ran 𝐹 ↔ 𝐹:𝑋–ontoβ†’βˆͺ (𝐾 β†Ύt ran 𝐹)))
1710, 16mpbid 231 . . . 4 (πœ‘ β†’ 𝐹:𝑋–ontoβ†’βˆͺ (𝐾 β†Ύt ran 𝐹))
18 toptopon2 22764 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
1912, 18sylib 217 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
20 ssidd 3998 . . . . . 6 (πœ‘ β†’ ran 𝐹 βŠ† ran 𝐹)
21 cnrest2 23134 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ ran 𝐹 βŠ† ran 𝐹 ∧ ran 𝐹 βŠ† βˆͺ 𝐾) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt ran 𝐹))))
2219, 20, 7, 21syl3anc 1368 . . . . 5 (πœ‘ β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt ran 𝐹))))
231, 22mpbid 231 . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt ran 𝐹)))
24 eqid 2724 . . . . 5 βˆͺ (𝐾 β†Ύt ran 𝐹) = βˆͺ (𝐾 β†Ύt ran 𝐹)
2524cnconn 23270 . . . 4 ((𝐽 ∈ Conn ∧ 𝐹:𝑋–ontoβ†’βˆͺ (𝐾 β†Ύt ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt ran 𝐹))) β†’ (𝐾 β†Ύt ran 𝐹) ∈ Conn)
268, 17, 23, 25syl3anc 1368 . . 3 (πœ‘ β†’ (𝐾 β†Ύt ran 𝐹) ∈ Conn)
27 conncn.u . . 3 (πœ‘ β†’ π‘ˆ ∈ 𝐾)
28 conncn.1 . . . 4 (πœ‘ β†’ (πΉβ€˜π΄) ∈ π‘ˆ)
29 conncn.a . . . . 5 (πœ‘ β†’ 𝐴 ∈ 𝑋)
30 fnfvelrn 7073 . . . . 5 ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ (πΉβ€˜π΄) ∈ ran 𝐹)
316, 29, 30syl2anc 583 . . . 4 (πœ‘ β†’ (πΉβ€˜π΄) ∈ ran 𝐹)
32 inelcm 4457 . . . 4 (((πΉβ€˜π΄) ∈ π‘ˆ ∧ (πΉβ€˜π΄) ∈ ran 𝐹) β†’ (π‘ˆ ∩ ran 𝐹) β‰  βˆ…)
3328, 31, 32syl2anc 583 . . 3 (πœ‘ β†’ (π‘ˆ ∩ ran 𝐹) β‰  βˆ…)
34 conncn.c . . 3 (πœ‘ β†’ π‘ˆ ∈ (Clsdβ€˜πΎ))
353, 7, 26, 27, 33, 34connsubclo 23272 . 2 (πœ‘ β†’ ran 𝐹 βŠ† π‘ˆ)
36 df-f 6538 . 2 (𝐹:π‘‹βŸΆπ‘ˆ ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹 βŠ† π‘ˆ))
376, 35, 36sylanbrc 582 1 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098   β‰  wne 2932   ∩ cin 3940   βŠ† wss 3941  βˆ…c0 4315  βˆͺ cuni 4900  ran crn 5668   Fn wfn 6529  βŸΆwf 6530  β€“ontoβ†’wfo 6532  β€˜cfv 6534  (class class class)co 7402   β†Ύt crest 17371  Topctop 22739  TopOnctopon 22756  Clsdccld 22864   Cn ccn 23072  Conncconn 23259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-map 8819  df-en 8937  df-fin 8940  df-fi 9403  df-rest 17373  df-topgen 17394  df-top 22740  df-topon 22757  df-bases 22793  df-cld 22867  df-cn 23075  df-conn 23260
This theorem is referenced by:  pconnconn  34740  cvmliftmolem1  34790  cvmlift2lem9  34820  cvmlift3lem6  34833
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