MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  concn Structured version   Unicode version

Theorem concn 19157
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 2452 . . . . 5  |-  U. K  =  U. K
42, 3cnf 18977 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
51, 4syl 16 . . 3  |-  ( ph  ->  F : X --> U. K
)
6 ffn 5662 . . 3  |-  ( F : X --> U. K  ->  F  Fn  X )
75, 6syl 16 . 2  |-  ( ph  ->  F  Fn  X )
8 frn 5668 . . . 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 5729 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
127, 11sylib 196 . . . . 5  |-  ( ph  ->  F : X -onto-> ran  F )
13 cntop2 18972 . . . . . . . 8  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
141, 13syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
153restuni 18893 . . . . . . 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 5721 . . . . . 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 18665 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2114, 20sylib 196 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
22 ssid 3478 . . . . . . 7  |-  ran  F  C_ 
ran  F
2322a1i 11 . . . . . 6  |-  ( ph  ->  ran  F  C_  ran  F )
24 cnrest2 19017 . . . . . 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 1219 . . . . 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 2452 . . . . 5  |-  U. ( Kt  ran  F )  =  U. ( Kt  ran  F )
2827cnconn 19153 . . . 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 1219 . . 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 5944 . . . . 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 3836 . . . 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 19155 . 2  |-  ( ph  ->  ran  F  C_  U
)
39 df-f 5525 . 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 1370    e. wcel 1758    =/= wne 2645    i^i cin 3430    C_ wss 3431   (/)c0 3740   U.cuni 4194   ran crn 4944    Fn wfn 5516   -->wf 5517   -onto->wfo 5519   ` cfv 5521  (class class class)co 6195   ↾t crest 14473   Topctop 18625  TopOnctopon 18626   Clsdccld 18747    Cn ccn 18955   Conccon 19142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-fin 7419  df-fi 7767  df-rest 14475  df-topgen 14496  df-top 18630  df-bases 18632  df-topon 18633  df-cld 18750  df-cn 18958  df-con 19143
This theorem is referenced by:  pconcon  27259  cvmliftmolem1  27309  cvmlift2lem9  27339  cvmlift3lem6  27352
  Copyright terms: Public domain W3C validator