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Theorem imacmp 17452
Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 18-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
imacmp  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Kt  ( F
" A ) )  e.  Comp )

Proof of Theorem imacmp
StepHypRef Expression
1 df-ima 4883 . . 3  |-  ( F
" A )  =  ran  ( F  |`  A )
21oveq2i 6084 . 2  |-  ( Kt  ( F " A ) )  =  ( Kt  ran  ( F  |`  A ) )
3 simpr 448 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  e.  Comp )
4 simpl 444 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  F  e.  ( J  Cn  K ) )
5 inss2 3554 . . . . 5  |-  ( A  i^i  U. J ) 
C_  U. J
6 eqid 2435 . . . . . 6  |-  U. J  =  U. J
76cnrest 17341 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( A  i^i  U. J
)  C_  U. J )  ->  ( F  |`  ( A  i^i  U. J
) )  e.  ( ( Jt  ( A  i^i  U. J ) )  Cn  K ) )
84, 5, 7sylancl 644 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  ( A  i^i  U. J
) )  e.  ( ( Jt  ( A  i^i  U. J ) )  Cn  K ) )
9 resdmres 5353 . . . . 5  |-  ( F  |`  dom  ( F  |`  A ) )  =  ( F  |`  A )
10 dmres 5159 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
11 eqid 2435 . . . . . . . . . 10  |-  U. K  =  U. K
126, 11cnf 17302 . . . . . . . . 9  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
13 fdm 5587 . . . . . . . . 9  |-  ( F : U. J --> U. K  ->  dom  F  =  U. J )
144, 12, 133syl 19 . . . . . . . 8  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  dom  F  =  U. J )
1514ineq2d 3534 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( A  i^i  dom 
F )  =  ( A  i^i  U. J
) )
1610, 15syl5eq 2479 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  dom  ( F  |`  A )  =  ( A  i^i  U. J
) )
1716reseq2d 5138 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  dom  ( F  |`  A ) )  =  ( F  |`  ( A  i^i  U. J ) ) )
189, 17syl5eqr 2481 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  =  ( F  |`  ( A  i^i  U. J ) ) )
19 cmptop 17450 . . . . . . 7  |-  ( ( Jt  A )  e.  Comp  -> 
( Jt  A )  e.  Top )
2019adantl 453 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  e.  Top )
21 restrcl 17213 . . . . . 6  |-  ( ( Jt  A )  e.  Top  ->  ( J  e.  _V  /\  A  e.  _V )
)
226restin 17222 . . . . . 6  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
2320, 21, 223syl 19 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
2423oveq1d 6088 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( ( Jt  A )  Cn  K )  =  ( ( Jt  ( A  i^i  U. J
) )  Cn  K
) )
258, 18, 243eltr4d 2516 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K
) )
26 rncmp 17451 . . 3  |-  ( ( ( Jt  A )  e.  Comp  /\  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )  ->  ( Kt  ran  ( F  |`  A ) )  e.  Comp )
273, 25, 26syl2anc 643 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Kt  ran  ( F  |`  A ) )  e.  Comp )
282, 27syl5eqel 2519 1  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Kt  ( F
" A ) )  e.  Comp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   U.cuni 4007   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   -->wf 5442  (class class class)co 6073   ↾t crest 13640   Topctop 16950    Cn ccn 17280   Compccmp 17441
This theorem is referenced by:  kgencn3  17582  txkgen  17676  xkoco1cn  17681  xkococnlem  17683  cmphaushmeo  17824  cnheiborlem  18971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-fin 7105  df-fi 7408  df-rest 13642  df-topgen 13659  df-top 16955  df-bases 16957  df-topon 16958  df-cn 17283  df-cmp 17442
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