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Theorem cnclima 18877
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 2443 . . . . . 6  |-  U. J  =  U. J
2 eqid 2443 . . . . . 6  |-  U. K  =  U. K
31, 2cnf 18855 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
43adantr 465 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  ->  F : U. J --> U. K
)
5 ffun 5566 . . . . . 6  |-  ( F : U. J --> U. K  ->  Fun  F )
6 funcnvcnv 5481 . . . . . 6  |-  ( Fun 
F  ->  Fun  `' `' F )
7 imadif 5498 . . . . . 6  |-  ( Fun  `' `' F  ->  ( `' F " ( U. K  \  A ) )  =  ( ( `' F " U. K
)  \  ( `' F " A ) ) )
85, 6, 73syl 20 . . . . 5  |-  ( F : U. J --> U. K  ->  ( `' F "
( U. K  \  A ) )  =  ( ( `' F " U. K )  \ 
( `' F " A ) ) )
9 fimacnv 5840 . . . . . 6  |-  ( F : U. J --> U. K  ->  ( `' F " U. K )  =  U. J )
109difeq1d 3478 . . . . 5  |-  ( F : U. J --> U. K  ->  ( ( `' F " U. K )  \ 
( `' F " A ) )  =  ( U. J  \ 
( `' F " A ) ) )
118, 10eqtr2d 2476 . . . 4  |-  ( F : U. J --> U. K  ->  ( U. J  \ 
( `' F " A ) )  =  ( `' F "
( U. K  \  A ) ) )
124, 11syl 16 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( U. J  \ 
( `' F " A ) )  =  ( `' F "
( U. K  \  A ) ) )
132cldopn 18640 . . . 4  |-  ( A  e.  ( Clsd `  K
)  ->  ( U. K  \  A )  e.  K )
14 cnima 18874 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( U. K  \  A
)  e.  K )  ->  ( `' F " ( U. K  \  A ) )  e.  J )
1513, 14sylan2 474 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( `' F "
( U. K  \  A ) )  e.  J )
1612, 15eqeltrd 2517 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( U. J  \ 
( `' F " A ) )  e.  J )
17 cntop1 18849 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
1817adantr 465 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  ->  J  e.  Top )
19 cnvimass 5194 . . . 4  |-  ( `' F " A ) 
C_  dom  F
20 fdm 5568 . . . . 5  |-  ( F : U. J --> U. K  ->  dom  F  =  U. J )
214, 20syl 16 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  ->  dom  F  =  U. J
)
2219, 21syl5sseq 3409 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( `' F " A )  C_  U. J
)
231iscld2 18637 . . 3  |-  ( ( J  e.  Top  /\  ( `' F " A ) 
C_  U. J )  -> 
( ( `' F " A )  e.  (
Clsd `  J )  <->  ( U. J  \  ( `' F " A ) )  e.  J ) )
2418, 22, 23syl2anc 661 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( ( `' F " A )  e.  (
Clsd `  J )  <->  ( U. J  \  ( `' F " A ) )  e.  J ) )
2516, 24mpbird 232 1  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( `' F " A )  e.  (
Clsd `  J )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3330    C_ wss 3333   U.cuni 4096   `'ccnv 4844   dom cdm 4845   "cima 4848   Fun wfun 5417   -->wf 5419   ` cfv 5423  (class class class)co 6096   Topctop 18503   Clsdccld 18625    Cn ccn 18833
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 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-top 18508  df-topon 18511  df-cld 18628  df-cn 18836
This theorem is referenced by:  iscncl  18878  cncls2i  18879  paste  18903  cnt1  18959  dnsconst  18987  cnconn  19031  hauseqlcld  19224  txcon  19267  imasncld  19269  r0cld  19316  kqreglem2  19320  kqnrmlem1  19321  kqnrmlem2  19322  hmeocld  19345  nrmhmph  19372  tgphaus  19692  csscld  20766  clsocv  20767  hmeoclda  28533  hmeocldb  28534  rfcnpre3  29760  rfcnpre4  29761
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