MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imacmp Unicode version

Theorem imacmp 17124
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 4702 . . 3  |-  ( F
" A )  =  ran  ( F  |`  A )
21oveq2i 5869 . 2  |-  ( Kt  ( F " A ) )  =  ( Kt  ran  ( F  |`  A ) )
3 simpr 447 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  e.  Comp )
4 resdmres 5164 . . . . . 6  |-  ( F  |`  dom  ( F  |`  A ) )  =  ( F  |`  A )
5 dmres 4976 . . . . . . . 8  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
6 simpl 443 . . . . . . . . . 10  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  F  e.  ( J  Cn  K ) )
7 eqid 2283 . . . . . . . . . . 11  |-  U. J  =  U. J
8 eqid 2283 . . . . . . . . . . 11  |-  U. K  =  U. K
97, 8cnf 16976 . . . . . . . . . 10  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
10 fdm 5393 . . . . . . . . . 10  |-  ( F : U. J --> U. K  ->  dom  F  =  U. J )
116, 9, 103syl 18 . . . . . . . . 9  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  dom  F  =  U. J )
1211ineq2d 3370 . . . . . . . 8  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( A  i^i  dom 
F )  =  ( A  i^i  U. J
) )
135, 12syl5eq 2327 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  dom  ( F  |`  A )  =  ( A  i^i  U. J
) )
1413reseq2d 4955 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  dom  ( F  |`  A ) )  =  ( F  |`  ( A  i^i  U. J ) ) )
154, 14syl5eqr 2329 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  =  ( F  |`  ( A  i^i  U. J ) ) )
16 inss2 3390 . . . . . 6  |-  ( A  i^i  U. J ) 
C_  U. J
177cnrest 17013 . . . . . 6  |-  ( ( 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 ) )
186, 16, 17sylancl 643 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  ( A  i^i  U. J
) )  e.  ( ( Jt  ( A  i^i  U. J ) )  Cn  K ) )
1915, 18eqeltrd 2357 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  e.  ( ( Jt  ( A  i^i  U. J ) )  Cn  K ) )
20 cmptop 17122 . . . . . . 7  |-  ( ( Jt  A )  e.  Comp  -> 
( Jt  A )  e.  Top )
2120adantl 452 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  e.  Top )
22 restrcl 16888 . . . . . 6  |-  ( ( Jt  A )  e.  Top  ->  ( J  e.  _V  /\  A  e.  _V )
)
237restin 16897 . . . . . 6  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
2421, 22, 233syl 18 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
2524oveq1d 5873 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( ( Jt  A )  Cn  K )  =  ( ( Jt  ( A  i^i  U. J
) )  Cn  K
) )
2619, 25eleqtrrd 2360 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K
) )
27 rncmp 17123 . . 3  |-  ( ( ( Jt  A )  e.  Comp  /\  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )  ->  ( Kt  ran  ( F  |`  A ) )  e.  Comp )
283, 26, 27syl2anc 642 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( Kt  ran  ( F  |`  A ) )  e.  Comp )
292, 28syl5eqel 2367 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   U.cuni 3827   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   -->wf 5251  (class class class)co 5858   ↾t crest 13325   Topctop 16631    Cn ccn 16954   Compccmp 17113
This theorem is referenced by:  kgencn3  17253  txkgen  17346  xkoco1cn  17351  xkococnlem  17353  cmphaushmeo  17491  cnheiborlem  18452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-cmp 17114
  Copyright terms: Public domain W3C validator