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Theorem conncn 22800
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 2733 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
42, 3cnf 22620 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
51, 4syl 17 . . 3 (πœ‘ β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
65ffnd 6673 . 2 (πœ‘ β†’ 𝐹 Fn 𝑋)
75frnd 6680 . . 3 (πœ‘ β†’ ran 𝐹 βŠ† βˆͺ 𝐾)
8 conncn.j . . . 4 (πœ‘ β†’ 𝐽 ∈ Conn)
9 dffn4 6766 . . . . . 6 (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–ontoβ†’ran 𝐹)
106, 9sylib 217 . . . . 5 (πœ‘ β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
11 cntop2 22615 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
121, 11syl 17 . . . . . . 7 (πœ‘ β†’ 𝐾 ∈ Top)
133restuni 22536 . . . . . . 7 ((𝐾 ∈ Top ∧ ran 𝐹 βŠ† βˆͺ 𝐾) β†’ ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹))
1412, 7, 13syl2anc 585 . . . . . 6 (πœ‘ β†’ ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹))
15 foeq3 6758 . . . . . 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 22290 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
1912, 18sylib 217 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
20 ssidd 3971 . . . . . 6 (πœ‘ β†’ ran 𝐹 βŠ† ran 𝐹)
21 cnrest2 22660 . . . . . 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 1372 . . . . 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 2733 . . . . 5 βˆͺ (𝐾 β†Ύt ran 𝐹) = βˆͺ (𝐾 β†Ύt ran 𝐹)
2524cnconn 22796 . . . 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 1372 . . 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 7035 . . . . 5 ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ (πΉβ€˜π΄) ∈ ran 𝐹)
316, 29, 30syl2anc 585 . . . 4 (πœ‘ β†’ (πΉβ€˜π΄) ∈ ran 𝐹)
32 inelcm 4428 . . . 4 (((πΉβ€˜π΄) ∈ π‘ˆ ∧ (πΉβ€˜π΄) ∈ ran 𝐹) β†’ (π‘ˆ ∩ ran 𝐹) β‰  βˆ…)
3328, 31, 32syl2anc 585 . . 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 22798 . 2 (πœ‘ β†’ ran 𝐹 βŠ† π‘ˆ)
36 df-f 6504 . 2 (𝐹:π‘‹βŸΆπ‘ˆ ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹 βŠ† π‘ˆ))
376, 35, 36sylanbrc 584 1 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286  βˆͺ cuni 4869  ran crn 5638   Fn wfn 6495  βŸΆwf 6496  β€“ontoβ†’wfo 6498  β€˜cfv 6500  (class class class)co 7361   β†Ύt crest 17310  Topctop 22265  TopOnctopon 22282  Clsdccld 22390   Cn ccn 22598  Conncconn 22785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-map 8773  df-en 8890  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cld 22393  df-cn 22601  df-conn 22786
This theorem is referenced by:  pconnconn  33889  cvmliftmolem1  33939  cvmlift2lem9  33969  cvmlift3lem6  33982
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