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Theorem conncn 22929
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 2732 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
42, 3cnf 22749 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
51, 4syl 17 . . 3 (πœ‘ β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
65ffnd 6718 . 2 (πœ‘ β†’ 𝐹 Fn 𝑋)
75frnd 6725 . . 3 (πœ‘ β†’ ran 𝐹 βŠ† βˆͺ 𝐾)
8 conncn.j . . . 4 (πœ‘ β†’ 𝐽 ∈ Conn)
9 dffn4 6811 . . . . . 6 (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–ontoβ†’ran 𝐹)
106, 9sylib 217 . . . . 5 (πœ‘ β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
11 cntop2 22744 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
121, 11syl 17 . . . . . . 7 (πœ‘ β†’ 𝐾 ∈ Top)
133restuni 22665 . . . . . . 7 ((𝐾 ∈ Top ∧ ran 𝐹 βŠ† βˆͺ 𝐾) β†’ ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹))
1412, 7, 13syl2anc 584 . . . . . 6 (πœ‘ β†’ ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹))
15 foeq3 6803 . . . . . 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 22419 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
1912, 18sylib 217 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
20 ssidd 4005 . . . . . 6 (πœ‘ β†’ ran 𝐹 βŠ† ran 𝐹)
21 cnrest2 22789 . . . . . 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 1371 . . . . 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 2732 . . . . 5 βˆͺ (𝐾 β†Ύt ran 𝐹) = βˆͺ (𝐾 β†Ύt ran 𝐹)
2524cnconn 22925 . . . 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 1371 . . 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 7082 . . . . 5 ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ (πΉβ€˜π΄) ∈ ran 𝐹)
316, 29, 30syl2anc 584 . . . 4 (πœ‘ β†’ (πΉβ€˜π΄) ∈ ran 𝐹)
32 inelcm 4464 . . . 4 (((πΉβ€˜π΄) ∈ π‘ˆ ∧ (πΉβ€˜π΄) ∈ ran 𝐹) β†’ (π‘ˆ ∩ ran 𝐹) β‰  βˆ…)
3328, 31, 32syl2anc 584 . . 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 22927 . 2 (πœ‘ β†’ ran 𝐹 βŠ† π‘ˆ)
36 df-f 6547 . 2 (𝐹:π‘‹βŸΆπ‘ˆ ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹 βŠ† π‘ˆ))
376, 35, 36sylanbrc 583 1 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  βˆͺ cuni 4908  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7408   β†Ύt crest 17365  Topctop 22394  TopOnctopon 22411  Clsdccld 22519   Cn ccn 22727  Conncconn 22914
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-map 8821  df-en 8939  df-fin 8942  df-fi 9405  df-rest 17367  df-topgen 17388  df-top 22395  df-topon 22412  df-bases 22448  df-cld 22522  df-cn 22730  df-conn 22915
This theorem is referenced by:  pconnconn  34217  cvmliftmolem1  34267  cvmlift2lem9  34297  cvmlift3lem6  34310
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