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

Theorem imacmp 19656
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 5005 . . 3  |-  ( F
" A )  =  ran  ( F  |`  A )
21oveq2i 6286 . 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 3712 . . . . 5  |-  ( A  i^i  U. J ) 
C_  U. J
6 eqid 2460 . . . . . 6  |-  U. J  =  U. J
76cnrest 19545 . . . . 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 5489 . . . . 5  |-  ( F  |`  dom  ( F  |`  A ) )  =  ( F  |`  A )
10 dmres 5285 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
11 eqid 2460 . . . . . . . . . 10  |-  U. K  =  U. K
126, 11cnf 19506 . . . . . . . . 9  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
13 fdm 5726 . . . . . . . . 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 3693 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( A  i^i  dom 
F )  =  ( A  i^i  U. J
) )
1610, 15syl5eq 2513 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  dom  ( F  |`  A )  =  ( A  i^i  U. J
) )
1716reseq2d 5264 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  dom  ( F  |`  A ) )  =  ( F  |`  ( A  i^i  U. J ) ) )
189, 17syl5eqr 2515 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  =  ( F  |`  ( A  i^i  U. J ) ) )
19 cmptop 19654 . . . . . . 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 19417 . . . . . 6  |-  ( ( Jt  A )  e.  Top  ->  ( J  e.  _V  /\  A  e.  _V )
)
226restin 19426 . . . . . 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 6290 . . . 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 2563 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( Jt  A )  e.  Comp )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K
) )
26 rncmp 19655 . . 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 2552 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 1374    e. wcel 1762   _Vcvv 3106    i^i cin 3468    C_ wss 3469   U.cuni 4238   dom cdm 4992   ran crn 4993    |` cres 4994   "cima 4995   -->wf 5575  (class class class)co 6275   ↾t crest 14665   Topctop 19154    Cn ccn 19484   Compccmp 19645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-fin 7510  df-fi 7860  df-rest 14667  df-topgen 14688  df-top 19159  df-bases 19161  df-topon 19162  df-cn 19487  df-cmp 19646
This theorem is referenced by:  kgencn3  19787  txkgen  19881  xkoco1cn  19886  xkococnlem  19888  cmphaushmeo  20029  cnheiborlem  21182
  Copyright terms: Public domain W3C validator