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Theorem concn 18989
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  = 
U. J
concn.j  |-  ( ph  ->  J  e.  Con )
concn.f  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
concn.u  |-  ( ph  ->  U  e.  K )
concn.c  |-  ( ph  ->  U  e.  ( Clsd `  K ) )
concn.a  |-  ( ph  ->  A  e.  X )
concn.1  |-  ( ph  ->  ( F `  A
)  e.  U )
Assertion
Ref Expression
concn  |-  ( ph  ->  F : X --> U )

Proof of Theorem concn
StepHypRef Expression
1 concn.f . . . 4  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
2 concn.x . . . . 5  |-  X  = 
U. J
3 eqid 2441 . . . . 5  |-  U. K  =  U. K
42, 3cnf 18809 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
51, 4syl 16 . . 3  |-  ( ph  ->  F : X --> U. K
)
6 ffn 5556 . . 3  |-  ( F : X --> U. K  ->  F  Fn  X )
75, 6syl 16 . 2  |-  ( ph  ->  F  Fn  X )
8 frn 5562 . . . 4  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
95, 8syl 16 . . 3  |-  ( ph  ->  ran  F  C_  U. K
)
10 concn.j . . . 4  |-  ( ph  ->  J  e.  Con )
11 dffn4 5623 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
127, 11sylib 196 . . . . 5  |-  ( ph  ->  F : X -onto-> ran  F )
13 cntop2 18804 . . . . . . . 8  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
141, 13syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
153restuni 18725 . . . . . . 7  |-  ( ( K  e.  Top  /\  ran  F  C_  U. K )  ->  ran  F  =  U. ( Kt  ran  F ) )
1614, 9, 15syl2anc 656 . . . . . 6  |-  ( ph  ->  ran  F  =  U. ( Kt  ran  F ) )
17 foeq3 5615 . . . . . 6  |-  ( ran 
F  =  U. ( Kt  ran  F )  ->  ( F : X -onto-> ran  F  <->  F : X -onto-> U. ( Kt  ran  F ) ) )
1816, 17syl 16 . . . . 5  |-  ( ph  ->  ( F : X -onto-> ran  F  <->  F : X -onto-> U. ( Kt  ran  F ) ) )
1912, 18mpbid 210 . . . 4  |-  ( ph  ->  F : X -onto-> U. ( Kt  ran  F ) )
203toptopon 18497 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2114, 20sylib 196 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
22 ssid 3372 . . . . . . 7  |-  ran  F  C_ 
ran  F
2322a1i 11 . . . . . 6  |-  ( ph  ->  ran  F  C_  ran  F )
24 cnrest2 18849 . . . . . 6  |-  ( ( K  e.  (TopOn `  U. K )  /\  ran  F 
C_  ran  F  /\  ran  F  C_  U. K )  ->  ( F  e.  ( J  Cn  K
)  <->  F  e.  ( J  Cn  ( Kt  ran  F
) ) ) )
2521, 23, 9, 24syl3anc 1213 . . . . 5  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
F  e.  ( J  Cn  ( Kt  ran  F
) ) ) )
261, 25mpbid 210 . . . 4  |-  ( ph  ->  F  e.  ( J  Cn  ( Kt  ran  F
) ) )
27 eqid 2441 . . . . 5  |-  U. ( Kt  ran  F )  =  U. ( Kt  ran  F )
2827cnconn 18985 . . . 4  |-  ( ( J  e.  Con  /\  F : X -onto-> U. ( Kt  ran  F )  /\  F  e.  ( J  Cn  ( Kt  ran  F ) ) )  ->  ( Kt  ran  F
)  e.  Con )
2910, 19, 26, 28syl3anc 1213 . . 3  |-  ( ph  ->  ( Kt  ran  F )  e. 
Con )
30 concn.u . . 3  |-  ( ph  ->  U  e.  K )
31 concn.1 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  U )
32 concn.a . . . . 5  |-  ( ph  ->  A  e.  X )
33 fnfvelrn 5837 . . . . 5  |-  ( ( F  Fn  X  /\  A  e.  X )  ->  ( F `  A
)  e.  ran  F
)
347, 32, 33syl2anc 656 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ran  F
)
35 inelcm 3730 . . . 4  |-  ( ( ( F `  A
)  e.  U  /\  ( F `  A )  e.  ran  F )  ->  ( U  i^i  ran 
F )  =/=  (/) )
3631, 34, 35syl2anc 656 . . 3  |-  ( ph  ->  ( U  i^i  ran  F )  =/=  (/) )
37 concn.c . . 3  |-  ( ph  ->  U  e.  ( Clsd `  K ) )
383, 9, 29, 30, 36, 37consubclo 18987 . 2  |-  ( ph  ->  ran  F  C_  U
)
39 df-f 5419 . 2  |-  ( F : X --> U  <->  ( F  Fn  X  /\  ran  F  C_  U ) )
407, 38, 39sylanbrc 659 1  |-  ( ph  ->  F : X --> U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1364    e. wcel 1761    =/= wne 2604    i^i cin 3324    C_ wss 3325   (/)c0 3634   U.cuni 4088   ran crn 4837    Fn wfn 5410   -->wf 5411   -onto->wfo 5413   ` cfv 5415  (class class class)co 6090   ↾t crest 14355   Topctop 18457  TopOnctopon 18458   Clsdccld 18579    Cn ccn 18787   Conccon 18974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-fin 7310  df-fi 7657  df-rest 14357  df-topgen 14378  df-top 18462  df-bases 18464  df-topon 18465  df-cld 18582  df-cn 18790  df-con 18975
This theorem is referenced by:  pconcon  27050  cvmliftmolem1  27100  cvmlift2lem9  27130  cvmlift3lem6  27143
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