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Theorem imacmp 19116
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 4951 . . 3  |-  ( F
" A )  =  ran  ( F  |`  A )
21oveq2i 6201 . 2  |-  ( Kt  ( F " A ) )  =  ( Kt  ran  ( F  |`  A ) )
3 simpr 461 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  e.  Comp )
4 simpl 457 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  F  e.  ( J  Cn  K ) )
5 inss2 3669 . . . . 5  |-  ( A  i^i  U. J ) 
C_  U. J
6 eqid 2451 . . . . . 6  |-  U. J  =  U. J
76cnrest 19005 . . . . 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 662 . . . 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 5427 . . . . 5  |-  ( F  |`  dom  ( F  |`  A ) )  =  ( F  |`  A )
10 dmres 5229 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
11 eqid 2451 . . . . . . . . . 10  |-  U. K  =  U. K
126, 11cnf 18966 . . . . . . . . 9  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
13 fdm 5661 . . . . . . . . 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 3650 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( A  i^i  dom 
F )  =  ( A  i^i  U. J
) )
1610, 15syl5eq 2504 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  dom  ( F  |`  A )  =  ( A  i^i  U. J
) )
1716reseq2d 5208 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  dom  ( F  |`  A ) )  =  ( F  |`  ( A  i^i  U. J ) ) )
189, 17syl5eqr 2506 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  =  ( F  |`  ( A  i^i  U. J ) ) )
19 cmptop 19114 . . . . . . 7  |-  ( ( Jt  A )  e.  Comp  -> 
( Jt  A )  e.  Top )
2019adantl 466 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  e.  Top )
21 restrcl 18877 . . . . . 6  |-  ( ( Jt  A )  e.  Top  ->  ( J  e.  _V  /\  A  e.  _V )
)
226restin 18886 . . . . . 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 6205 . . . 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 2554 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K
) )
26 rncmp 19115 . . 3  |-  ( ( ( Jt  A )  e.  Comp  /\  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )  ->  ( Kt  ran  ( F  |`  A ) )  e.  Comp )
273, 25, 26syl2anc 661 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Kt  ran  ( F  |`  A ) )  e.  Comp )
282, 27syl5eqel 2543 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3068    i^i cin 3425    C_ wss 3426   U.cuni 4189   dom cdm 4938   ran crn 4939    |` cres 4940   "cima 4941   -->wf 5512  (class class class)co 6190   ↾t crest 14461   Topctop 18614    Cn ccn 18944   Compccmp 19105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-fin 7414  df-fi 7762  df-rest 14463  df-topgen 14484  df-top 18619  df-bases 18621  df-topon 18622  df-cn 18947  df-cmp 19106
This theorem is referenced by:  kgencn3  19247  txkgen  19341  xkoco1cn  19346  xkococnlem  19348  cmphaushmeo  19489  cnheiborlem  20642
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