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Theorem conncn 23343
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 2728 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
42, 3cnf 23163 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
51, 4syl 17 . . 3 (πœ‘ β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
65ffnd 6723 . 2 (πœ‘ β†’ 𝐹 Fn 𝑋)
75frnd 6730 . . 3 (πœ‘ β†’ ran 𝐹 βŠ† βˆͺ 𝐾)
8 conncn.j . . . 4 (πœ‘ β†’ 𝐽 ∈ Conn)
9 dffn4 6817 . . . . . 6 (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–ontoβ†’ran 𝐹)
106, 9sylib 217 . . . . 5 (πœ‘ β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
11 cntop2 23158 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
121, 11syl 17 . . . . . . 7 (πœ‘ β†’ 𝐾 ∈ Top)
133restuni 23079 . . . . . . 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 6809 . . . . . 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 22833 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
1912, 18sylib 217 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
20 ssidd 4003 . . . . . 6 (πœ‘ β†’ ran 𝐹 βŠ† ran 𝐹)
21 cnrest2 23203 . . . . . 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 1369 . . . . 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 2728 . . . . 5 βˆͺ (𝐾 β†Ύt ran 𝐹) = βˆͺ (𝐾 β†Ύt ran 𝐹)
2524cnconn 23339 . . . 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 1369 . . 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 7090 . . . . 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 4465 . . . 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 23341 . 2 (πœ‘ β†’ ran 𝐹 βŠ† π‘ˆ)
36 df-f 6552 . 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 1534   ∈ wcel 2099   β‰  wne 2937   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4323  βˆͺ cuni 4908  ran crn 5679   Fn wfn 6543  βŸΆwf 6544  β€“ontoβ†’wfo 6546  β€˜cfv 6548  (class class class)co 7420   β†Ύt crest 17402  Topctop 22808  TopOnctopon 22825  Clsdccld 22933   Cn ccn 23141  Conncconn 23328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-map 8847  df-en 8965  df-fin 8968  df-fi 9435  df-rest 17404  df-topgen 17425  df-top 22809  df-topon 22826  df-bases 22862  df-cld 22936  df-cn 23144  df-conn 23329
This theorem is referenced by:  pconnconn  34841  cvmliftmolem1  34891  cvmlift2lem9  34921  cvmlift3lem6  34934
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