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Theorem rncmp 19764
Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
rncmp  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( Kt  ran  F )  e.  Comp )

Proof of Theorem rncmp
StepHypRef Expression
1 simpl 457 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  J  e.  Comp )
2 eqid 2467 . . . . . . 7  |-  U. J  =  U. J
3 eqid 2467 . . . . . . 7  |-  U. K  =  U. K
42, 3cnf 19615 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
54adantl 466 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F : U. J --> U. K
)
6 ffn 5737 . . . . 5  |-  ( F : U. J --> U. K  ->  F  Fn  U. J
)
75, 6syl 16 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F  Fn  U. J )
8 dffn4 5807 . . . 4  |-  ( F  Fn  U. J  <->  F : U. J -onto-> ran  F )
97, 8sylib 196 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F : U. J -onto-> ran  F
)
10 cntop2 19610 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
1110adantl 466 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  K  e.  Top )
12 frn 5743 . . . . . 6  |-  ( F : U. J --> U. K  ->  ran  F  C_  U. K
)
135, 12syl 16 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ran  F 
C_  U. K )
143restuni 19531 . . . . 5  |-  ( ( K  e.  Top  /\  ran  F  C_  U. K )  ->  ran  F  =  U. ( Kt  ran  F ) )
1511, 13, 14syl2anc 661 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ran  F  =  U. ( Kt  ran 
F ) )
16 foeq3 5799 . . . 4  |-  ( ran 
F  =  U. ( Kt  ran  F )  ->  ( F : U. J -onto-> ran  F  <-> 
F : U. J -onto-> U. ( Kt  ran  F ) ) )
1715, 16syl 16 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( F : U. J -onto-> ran  F  <-> 
F : U. J -onto-> U. ( Kt  ran  F ) ) )
189, 17mpbid 210 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F : U. J -onto-> U. ( Kt  ran  F ) )
19 simpr 461 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F  e.  ( J  Cn  K
) )
203toptopon 19303 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2111, 20sylib 196 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  K  e.  (TopOn `  U. K ) )
22 ssid 3528 . . . . 5  |-  ran  F  C_ 
ran  F
2322a1i 11 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ran  F 
C_  ran  F )
24 cnrest2 19655 . . . 4  |-  ( ( K  e.  (TopOn `  U. K )  /\  ran  F 
C_  ran  F  /\  ran  F  C_  U. K )  ->  ( F  e.  ( J  Cn  K
)  <->  F  e.  ( J  Cn  ( Kt  ran  F
) ) ) )
2521, 23, 13, 24syl3anc 1228 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( F  e.  ( J  Cn  K )  <->  F  e.  ( J  Cn  ( Kt  ran  F ) ) ) )
2619, 25mpbid 210 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F  e.  ( J  Cn  ( Kt  ran  F ) ) )
27 eqid 2467 . . 3  |-  U. ( Kt  ran  F )  =  U. ( Kt  ran  F )
2827cncmp 19760 . 2  |-  ( ( J  e.  Comp  /\  F : U. J -onto-> U. ( Kt  ran  F )  /\  F  e.  ( J  Cn  ( Kt  ran  F ) ) )  ->  ( Kt  ran  F
)  e.  Comp )
291, 18, 26, 28syl3anc 1228 1  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( Kt  ran  F )  e.  Comp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   U.cuni 4251   ran crn 5006    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295   ↾t crest 14693   Topctop 19263  TopOnctopon 19264    Cn ccn 19593   Compccmp 19754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-fin 7532  df-fi 7883  df-rest 14695  df-topgen 14716  df-top 19268  df-bases 19270  df-topon 19271  df-cn 19596  df-cmp 19755
This theorem is referenced by:  imacmp  19765  kgencn2  19926  bndth  21326
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