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Theorem cnresima 28606
Description: A continuous function is continuous onto its image. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
Assertion
Ref Expression
cnresima  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Cn  K
) )  ->  F  e.  ( J  Cn  ( Kt  ran  F ) ) )

Proof of Theorem cnresima
StepHypRef Expression
1 simp3 990 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Cn  K
) )  ->  F  e.  ( J  Cn  K
) )
2 simp2 989 . . . 4  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Cn  K
) )  ->  K  e.  Top )
3 eqid 2437 . . . . 5  |-  U. K  =  U. K
43toptopon 18507 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
52, 4sylib 196 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Cn  K
) )  ->  K  e.  (TopOn `  U. K ) )
6 ssid 3368 . . . 4  |-  ran  F  C_ 
ran  F
76a1i 11 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Cn  K
) )  ->  ran  F 
C_  ran  F )
8 eqid 2437 . . . . . 6  |-  U. J  =  U. J
98, 3cnf 18819 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
10 frn 5558 . . . . 5  |-  ( F : U. J --> U. K  ->  ran  F  C_  U. K
)
119, 10syl 16 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  ran  F 
C_  U. K )
12113ad2ant3 1011 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Cn  K
) )  ->  ran  F 
C_  U. K )
13 cnrest2 18859 . . 3  |-  ( ( 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
) ) ) )
145, 7, 12, 13syl3anc 1218 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Cn  K
) )  ->  ( F  e.  ( J  Cn  K )  <->  F  e.  ( J  Cn  ( Kt  ran  F ) ) ) )
151, 14mpbid 210 1  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Cn  K
) )  ->  F  e.  ( J  Cn  ( Kt  ran  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    e. wcel 1756    C_ wss 3321   U.cuni 4084   ran crn 4833   -->wf 5407   ` cfv 5411  (class class class)co 6086   ↾t crest 14351   Topctop 18467  TopOnctopon 18468    Cn ccn 18797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-fin 7306  df-fi 7653  df-rest 14353  df-topgen 14374  df-top 18472  df-bases 18474  df-topon 18475  df-cn 18800
This theorem is referenced by: (None)
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