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Theorem concn 19721
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 2467 . . . . 5  |-  U. K  =  U. K
42, 3cnf 19541 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
51, 4syl 16 . . 3  |-  ( ph  ->  F : X --> U. K
)
6 ffn 5731 . . 3  |-  ( F : X --> U. K  ->  F  Fn  X )
75, 6syl 16 . 2  |-  ( ph  ->  F  Fn  X )
8 frn 5737 . . . 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 5801 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
127, 11sylib 196 . . . . 5  |-  ( ph  ->  F : X -onto-> ran  F )
13 cntop2 19536 . . . . . . . 8  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
141, 13syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
153restuni 19457 . . . . . . 7  |-  ( ( K  e.  Top  /\  ran  F  C_  U. K )  ->  ran  F  =  U. ( Kt  ran  F ) )
1614, 9, 15syl2anc 661 . . . . . 6  |-  ( ph  ->  ran  F  =  U. ( Kt  ran  F ) )
17 foeq3 5793 . . . . . 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 19229 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2114, 20sylib 196 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
22 ssid 3523 . . . . . . 7  |-  ran  F  C_ 
ran  F
2322a1i 11 . . . . . 6  |-  ( ph  ->  ran  F  C_  ran  F )
24 cnrest2 19581 . . . . . 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 1228 . . . . 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 2467 . . . . 5  |-  U. ( Kt  ran  F )  =  U. ( Kt  ran  F )
2827cnconn 19717 . . . 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 1228 . . 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 6018 . . . . 5  |-  ( ( F  Fn  X  /\  A  e.  X )  ->  ( F `  A
)  e.  ran  F
)
347, 32, 33syl2anc 661 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ran  F
)
35 inelcm 3881 . . . 4  |-  ( ( ( F `  A
)  e.  U  /\  ( F `  A )  e.  ran  F )  ->  ( U  i^i  ran 
F )  =/=  (/) )
3631, 34, 35syl2anc 661 . . 3  |-  ( ph  ->  ( U  i^i  ran  F )  =/=  (/) )
37 concn.c . . 3  |-  ( ph  ->  U  e.  ( Clsd `  K ) )
383, 9, 29, 30, 36, 37consubclo 19719 . 2  |-  ( ph  ->  ran  F  C_  U
)
39 df-f 5592 . 2  |-  ( F : X --> U  <->  ( F  Fn  X  /\  ran  F  C_  U ) )
407, 38, 39sylanbrc 664 1  |-  ( ph  ->  F : X --> U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662    i^i cin 3475    C_ wss 3476   (/)c0 3785   U.cuni 4245   ran crn 5000    Fn wfn 5583   -->wf 5584   -onto->wfo 5586   ` cfv 5588  (class class class)co 6284   ↾t crest 14676   Topctop 19189  TopOnctopon 19190   Clsdccld 19311    Cn ccn 19519   Conccon 19706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-fin 7520  df-fi 7871  df-rest 14678  df-topgen 14699  df-top 19194  df-bases 19196  df-topon 19197  df-cld 19314  df-cn 19522  df-con 19707
This theorem is referenced by:  pconcon  28344  cvmliftmolem1  28394  cvmlift2lem9  28424  cvmlift3lem6  28437
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