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Theorem concn 20518
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 2471 . . . . 5  |-  U. K  =  U. K
42, 3cnf 20339 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
51, 4syl 17 . . 3  |-  ( ph  ->  F : X --> U. K
)
6 ffn 5739 . . 3  |-  ( F : X --> U. K  ->  F  Fn  X )
75, 6syl 17 . 2  |-  ( ph  ->  F  Fn  X )
8 frn 5747 . . . 4  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
95, 8syl 17 . . 3  |-  ( ph  ->  ran  F  C_  U. K
)
10 concn.j . . . 4  |-  ( ph  ->  J  e.  Con )
11 dffn4 5812 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
127, 11sylib 201 . . . . 5  |-  ( ph  ->  F : X -onto-> ran  F )
13 cntop2 20334 . . . . . . . 8  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
141, 13syl 17 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
153restuni 20255 . . . . . . 7  |-  ( ( K  e.  Top  /\  ran  F  C_  U. K )  ->  ran  F  =  U. ( Kt  ran  F ) )
1614, 9, 15syl2anc 673 . . . . . 6  |-  ( ph  ->  ran  F  =  U. ( Kt  ran  F ) )
17 foeq3 5804 . . . . . 6  |-  ( ran 
F  =  U. ( Kt  ran  F )  ->  ( F : X -onto-> ran  F  <->  F : X -onto-> U. ( Kt  ran  F ) ) )
1816, 17syl 17 . . . . 5  |-  ( ph  ->  ( F : X -onto-> ran  F  <->  F : X -onto-> U. ( Kt  ran  F ) ) )
1912, 18mpbid 215 . . . 4  |-  ( ph  ->  F : X -onto-> U. ( Kt  ran  F ) )
203toptopon 20025 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2114, 20sylib 201 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
22 ssid 3437 . . . . . . 7  |-  ran  F  C_ 
ran  F
2322a1i 11 . . . . . 6  |-  ( ph  ->  ran  F  C_  ran  F )
24 cnrest2 20379 . . . . . 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 1292 . . . . 5  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
F  e.  ( J  Cn  ( Kt  ran  F
) ) ) )
261, 25mpbid 215 . . . 4  |-  ( ph  ->  F  e.  ( J  Cn  ( Kt  ran  F
) ) )
27 eqid 2471 . . . . 5  |-  U. ( Kt  ran  F )  =  U. ( Kt  ran  F )
2827cnconn 20514 . . . 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 1292 . . 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 6034 . . . . 5  |-  ( ( F  Fn  X  /\  A  e.  X )  ->  ( F `  A
)  e.  ran  F
)
347, 32, 33syl2anc 673 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ran  F
)
35 inelcm 3823 . . . 4  |-  ( ( ( F `  A
)  e.  U  /\  ( F `  A )  e.  ran  F )  ->  ( U  i^i  ran 
F )  =/=  (/) )
3631, 34, 35syl2anc 673 . . 3  |-  ( ph  ->  ( U  i^i  ran  F )  =/=  (/) )
37 concn.c . . 3  |-  ( ph  ->  U  e.  ( Clsd `  K ) )
383, 9, 29, 30, 36, 37consubclo 20516 . 2  |-  ( ph  ->  ran  F  C_  U
)
39 df-f 5593 . 2  |-  ( F : X --> U  <->  ( F  Fn  X  /\  ran  F  C_  U ) )
407, 38, 39sylanbrc 677 1  |-  ( ph  ->  F : X --> U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904    =/= wne 2641    i^i cin 3389    C_ wss 3390   (/)c0 3722   U.cuni 4190   ran crn 4840    Fn wfn 5584   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   ↾t crest 15397   Topctop 19994  TopOnctopon 19995   Clsdccld 20108    Cn ccn 20317   Conccon 20503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cld 20111  df-cn 20320  df-con 20504
This theorem is referenced by:  pconcon  30026  cvmliftmolem1  30076  cvmlift2lem9  30106  cvmlift3lem6  30119
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