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Theorem cnclima 19939
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 2454 . . . . . 6  |-  U. J  =  U. J
2 eqid 2454 . . . . . 6  |-  U. K  =  U. K
31, 2cnf 19917 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
43adantr 463 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  ->  F : U. J --> U. K
)
5 ffun 5715 . . . . . 6  |-  ( F : U. J --> U. K  ->  Fun  F )
6 funcnvcnv 5628 . . . . . 6  |-  ( Fun 
F  ->  Fun  `' `' F )
7 imadif 5645 . . . . . 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 5995 . . . . . 6  |-  ( F : U. J --> U. K  ->  ( `' F " U. K )  =  U. J )
109difeq1d 3607 . . . . 5  |-  ( F : U. J --> U. K  ->  ( ( `' F " U. K )  \ 
( `' F " A ) )  =  ( U. J  \ 
( `' F " A ) ) )
118, 10eqtr2d 2496 . . . 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 19702 . . . 4  |-  ( A  e.  ( Clsd `  K
)  ->  ( U. K  \  A )  e.  K )
14 cnima 19936 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( U. K  \  A
)  e.  K )  ->  ( `' F " ( U. K  \  A ) )  e.  J )
1513, 14sylan2 472 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( `' F "
( U. K  \  A ) )  e.  J )
1612, 15eqeltrd 2542 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( U. J  \ 
( `' F " A ) )  e.  J )
17 cntop1 19911 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
1817adantr 463 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  ->  J  e.  Top )
19 cnvimass 5345 . . . 4  |-  ( `' F " A ) 
C_  dom  F
20 fdm 5717 . . . . 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 3537 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  ( Clsd `  K ) )  -> 
( `' F " A )  C_  U. J
)
231iscld2 19699 . . 3  |-  ( ( J  e.  Top  /\  ( `' F " A ) 
C_  U. J )  -> 
( ( `' F " A )  e.  (
Clsd `  J )  <->  ( U. J  \  ( `' F " A ) )  e.  J ) )
2418, 22, 23syl2anc 659 . 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 367    = wceq 1398    e. wcel 1823    \ cdif 3458    C_ wss 3461   U.cuni 4235   `'ccnv 4987   dom cdm 4988   "cima 4991   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270   Topctop 19564   Clsdccld 19687    Cn ccn 19895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-top 19569  df-topon 19572  df-cld 19690  df-cn 19898
This theorem is referenced by:  iscncl  19940  cncls2i  19941  paste  19965  cnt1  20021  dnsconst  20049  cnconn  20092  hauseqlcld  20316  txcon  20359  imasncld  20361  r0cld  20408  kqreglem2  20412  kqnrmlem1  20413  kqnrmlem2  20414  hmeocld  20437  nrmhmph  20464  tgphaus  20784  csscld  21858  clsocv  21859  hmeoclda  30394  hmeocldb  30395  rfcnpre3  31651  rfcnpre4  31652
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