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Theorem cnclima 12173
Description: A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnclima  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( `' F " A )  e.  (
Clsd `  J )
)

Proof of Theorem cnclima
StepHypRef Expression
1 eqid 2100 . . . . . 6  |-  U. J  =  U. J
2 eqid 2100 . . . . . 6  |-  U. K  =  U. K
31, 2cnf 12154 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
43adantr 272 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  ->  F : U. J --> U. K
)
5 ffun 5211 . . . . . 6  |-  ( F : U. J --> U. K  ->  Fun  F )
6 funcnvcnv 5118 . . . . . 6  |-  ( Fun 
F  ->  Fun  `' `' F )
7 imadif 5139 . . . . . 6  |-  ( Fun  `' `' F  ->  ( `' F " ( U. K  \  A ) )  =  ( ( `' F " U. K
)  \  ( `' F " A ) ) )
85, 6, 73syl 17 . . . . 5  |-  ( F : U. J --> U. K  ->  ( `' F "
( U. K  \  A ) )  =  ( ( `' F " U. K )  \ 
( `' F " A ) ) )
9 fimacnv 5481 . . . . . 6  |-  ( F : U. J --> U. K  ->  ( `' F " U. K )  =  U. J )
109difeq1d 3140 . . . . 5  |-  ( F : U. J --> U. K  ->  ( ( `' F " U. K )  \ 
( `' F " A ) )  =  ( U. J  \ 
( `' F " A ) ) )
118, 10eqtr2d 2133 . . . 4  |-  ( F : U. J --> U. K  ->  ( U. J  \ 
( `' F " A ) )  =  ( `' F "
( U. K  \  A ) ) )
124, 11syl 14 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( U. J  \ 
( `' F " A ) )  =  ( `' F "
( U. K  \  A ) ) )
132cldopn 12058 . . . 4  |-  ( A  e.  ( Clsd `  K
)  ->  ( U. K  \  A )  e.  K )
14 cnima 12170 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( U. K  \  A
)  e.  K )  ->  ( `' F " ( U. K  \  A ) )  e.  J )
1513, 14sylan2 282 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( `' F "
( U. K  \  A ) )  e.  J )
1612, 15eqeltrd 2176 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( U. J  \ 
( `' F " A ) )  e.  J )
17 cntop1 12151 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
1817adantr 272 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  ->  J  e.  Top )
19 cnvimass 4838 . . . 4  |-  ( `' F " A ) 
C_  dom  F
2019, 4fssdm 5223 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( `' F " A )  C_  U. J
)
211iscld2 12055 . . 3  |-  ( ( J  e.  Top  /\  ( `' F " A ) 
C_  U. J )  -> 
( ( `' F " A )  e.  (
Clsd `  J )  <->  ( U. J  \  ( `' F " A ) )  e.  J ) )
2218, 20, 21syl2anc 406 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( ( `' F " A )  e.  (
Clsd `  J )  <->  ( U. J  \  ( `' F " A ) )  e.  J ) )
2316, 22mpbird 166 1  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( `' F " A )  e.  (
Clsd `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448    \ cdif 3018    C_ wss 3021   U.cuni 3683   `'ccnv 4476   "cima 4480   Fun wfun 5053   -->wf 5055   ` cfv 5059  (class class class)co 5706   Topctop 11946   Clsdccld 12043    Cn ccn 12136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-map 6474  df-top 11947  df-topon 11960  df-cld 12046  df-cn 12139
This theorem is referenced by: (None)
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