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Theorem imacmp 20064
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 5001 . . 3  |-  ( F
" A )  =  ran  ( F  |`  A )
21oveq2i 6281 . 2  |-  ( Kt  ( F " A ) )  =  ( Kt  ran  ( F  |`  A ) )
3 simpr 459 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  e.  Comp )
4 simpl 455 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  F  e.  ( J  Cn  K ) )
5 inss2 3705 . . . . 5  |-  ( A  i^i  U. J ) 
C_  U. J
6 eqid 2454 . . . . . 6  |-  U. J  =  U. J
76cnrest 19953 . . . . 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 660 . . . 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 5481 . . . . 5  |-  ( F  |`  dom  ( F  |`  A ) )  =  ( F  |`  A )
10 dmres 5282 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
11 eqid 2454 . . . . . . . . . 10  |-  U. K  =  U. K
126, 11cnf 19914 . . . . . . . . 9  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
13 fdm 5717 . . . . . . . . 9  |-  ( F : U. J --> U. K  ->  dom  F  =  U. J )
144, 12, 133syl 20 . . . . . . . 8  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  dom  F  =  U. J )
1514ineq2d 3686 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( A  i^i  dom 
F )  =  ( A  i^i  U. J
) )
1610, 15syl5eq 2507 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  dom  ( F  |`  A )  =  ( A  i^i  U. J
) )
1716reseq2d 5262 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  dom  ( F  |`  A ) )  =  ( F  |`  ( A  i^i  U. J ) ) )
189, 17syl5eqr 2509 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  =  ( F  |`  ( A  i^i  U. J ) ) )
19 cmptop 20062 . . . . . . 7  |-  ( ( Jt  A )  e.  Comp  -> 
( Jt  A )  e.  Top )
2019adantl 464 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  e.  Top )
21 restrcl 19825 . . . . . 6  |-  ( ( Jt  A )  e.  Top  ->  ( J  e.  _V  /\  A  e.  _V )
)
226restin 19834 . . . . . 6  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
2320, 21, 223syl 20 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
2423oveq1d 6285 . . . 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 2557 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K
) )
26 rncmp 20063 . . 3  |-  ( ( ( Jt  A )  e.  Comp  /\  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )  ->  ( Kt  ran  ( F  |`  A ) )  e.  Comp )
273, 25, 26syl2anc 659 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Kt  ran  ( F  |`  A ) )  e.  Comp )
282, 27syl5eqel 2546 1  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Kt  ( F
" A ) )  e.  Comp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    i^i cin 3460    C_ wss 3461   U.cuni 4235   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991   -->wf 5566  (class class class)co 6270   ↾t crest 14910   Topctop 19561    Cn ccn 19892   Compccmp 20053
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-rep 4550  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-3or 972  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-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  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-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-fin 7513  df-fi 7863  df-rest 14912  df-topgen 14933  df-top 19566  df-bases 19568  df-topon 19569  df-cn 19895  df-cmp 20054
This theorem is referenced by:  kgencn3  20225  txkgen  20319  xkoco1cn  20324  xkococnlem  20326  cmphaushmeo  20467  cnheiborlem  21620
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