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Theorem rncmp 19141
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 2454 . . . . . . 7  |-  U. J  =  U. J
3 eqid 2454 . . . . . . 7  |-  U. K  =  U. K
42, 3cnf 18992 . . . . . 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 5670 . . . . 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 5737 . . . 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 18987 . . . . . 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 5676 . . . . . 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 18908 . . . . 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 5729 . . . 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 18680 . . . . 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 3486 . . . . 5  |-  ran  F  C_ 
ran  F
2322a1i 11 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ran  F 
C_  ran  F )
24 cnrest2 19032 . . . 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 1219 . . 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 2454 . . 3  |-  U. ( Kt  ran  F )  =  U. ( Kt  ran  F )
2827cncmp 19137 . 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 1219 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 1370    e. wcel 1758    C_ wss 3439   U.cuni 4202   ran crn 4952    Fn wfn 5524   -->wf 5525   -onto->wfo 5527   ` cfv 5529  (class class class)co 6203   ↾t crest 14482   Topctop 18640  TopOnctopon 18641    Cn ccn 18970   Compccmp 19131
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-fin 7427  df-fi 7776  df-rest 14484  df-topgen 14505  df-top 18645  df-bases 18647  df-topon 18648  df-cn 18973  df-cmp 19132
This theorem is referenced by:  imacmp  19142  kgencn2  19272  bndth  20672
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