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Theorem concn 20372
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 2429 . . . . 5  |-  U. K  =  U. K
42, 3cnf 20193 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
51, 4syl 17 . . 3  |-  ( ph  ->  F : X --> U. K
)
6 ffn 5746 . . 3  |-  ( F : X --> U. K  ->  F  Fn  X )
75, 6syl 17 . 2  |-  ( ph  ->  F  Fn  X )
8 frn 5752 . . . 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 5816 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
127, 11sylib 199 . . . . 5  |-  ( ph  ->  F : X -onto-> ran  F )
13 cntop2 20188 . . . . . . . 8  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
141, 13syl 17 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
153restuni 20109 . . . . . . 7  |-  ( ( K  e.  Top  /\  ran  F  C_  U. K )  ->  ran  F  =  U. ( Kt  ran  F ) )
1614, 9, 15syl2anc 665 . . . . . 6  |-  ( ph  ->  ran  F  =  U. ( Kt  ran  F ) )
17 foeq3 5808 . . . . . 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 213 . . . 4  |-  ( ph  ->  F : X -onto-> U. ( Kt  ran  F ) )
203toptopon 19879 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2114, 20sylib 199 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
22 ssid 3489 . . . . . . 7  |-  ran  F  C_ 
ran  F
2322a1i 11 . . . . . 6  |-  ( ph  ->  ran  F  C_  ran  F )
24 cnrest2 20233 . . . . . 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 1264 . . . . 5  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
F  e.  ( J  Cn  ( Kt  ran  F
) ) ) )
261, 25mpbid 213 . . . 4  |-  ( ph  ->  F  e.  ( J  Cn  ( Kt  ran  F
) ) )
27 eqid 2429 . . . . 5  |-  U. ( Kt  ran  F )  =  U. ( Kt  ran  F )
2827cnconn 20368 . . . 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 1264 . . 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 665 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ran  F
)
35 inelcm 3853 . . . 4  |-  ( ( ( F `  A
)  e.  U  /\  ( F `  A )  e.  ran  F )  ->  ( U  i^i  ran 
F )  =/=  (/) )
3631, 34, 35syl2anc 665 . . 3  |-  ( ph  ->  ( U  i^i  ran  F )  =/=  (/) )
37 concn.c . . 3  |-  ( ph  ->  U  e.  ( Clsd `  K ) )
383, 9, 29, 30, 36, 37consubclo 20370 . 2  |-  ( ph  ->  ran  F  C_  U
)
39 df-f 5605 . 2  |-  ( F : X --> U  <->  ( F  Fn  X  /\  ran  F  C_  U ) )
407, 38, 39sylanbrc 668 1  |-  ( ph  ->  F : X --> U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1870    =/= wne 2625    i^i cin 3441    C_ wss 3442   (/)c0 3767   U.cuni 4222   ran crn 4855    Fn wfn 5596   -->wf 5597   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305   ↾t crest 15278   Topctop 19848  TopOnctopon 19849   Clsdccld 19962    Cn ccn 20171   Conccon 20357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-fin 7581  df-fi 7931  df-rest 15280  df-topgen 15301  df-top 19852  df-bases 19853  df-topon 19854  df-cld 19965  df-cn 20174  df-con 20358
This theorem is referenced by:  pconcon  29742  cvmliftmolem1  29792  cvmlift2lem9  29822  cvmlift3lem6  29835
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