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Theorem rncmp 17139
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 443 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  J  e.  Comp )
2 eqid 2296 . . . . . . 7  |-  U. J  =  U. J
3 eqid 2296 . . . . . . 7  |-  U. K  =  U. K
42, 3cnf 16992 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
54adantl 452 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F : U. J --> U. K
)
6 ffn 5405 . . . . 5  |-  ( F : U. J --> U. K  ->  F  Fn  U. J
)
75, 6syl 15 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F  Fn  U. J )
8 dffn4 5473 . . . 4  |-  ( F  Fn  U. J  <->  F : U. J -onto-> ran  F )
97, 8sylib 188 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F : U. J -onto-> ran  F
)
10 cntop2 16987 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
1110adantl 452 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  K  e.  Top )
12 frn 5411 . . . . . 6  |-  ( F : U. J --> U. K  ->  ran  F  C_  U. K
)
135, 12syl 15 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ran  F 
C_  U. K )
143restuni 16909 . . . . 5  |-  ( ( K  e.  Top  /\  ran  F  C_  U. K )  ->  ran  F  =  U. ( Kt  ran  F ) )
1511, 13, 14syl2anc 642 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ran  F  =  U. ( Kt  ran 
F ) )
16 foeq3 5465 . . . 4  |-  ( ran 
F  =  U. ( Kt  ran  F )  ->  ( F : U. J -onto-> ran  F  <-> 
F : U. J -onto-> U. ( Kt  ran  F ) ) )
1715, 16syl 15 . . 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 201 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F : U. J -onto-> U. ( Kt  ran  F ) )
19 simpr 447 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F  e.  ( J  Cn  K
) )
203toptopon 16687 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2111, 20sylib 188 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  K  e.  (TopOn `  U. K ) )
22 ssid 3210 . . . . 5  |-  ran  F  C_ 
ran  F
2322a1i 10 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ran  F 
C_  ran  F )
24 cnrest2 17030 . . . 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 1182 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( F  e.  ( J  Cn  K )  <->  F  e.  ( J  Cn  ( Kt  ran  F ) ) ) )
2619, 25mpbid 201 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  F  e.  ( J  Cn  ( Kt  ran  F ) ) )
27 eqid 2296 . . 3  |-  U. ( Kt  ran  F )  =  U. ( Kt  ran  F )
2827cncmp 17135 . 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 1182 1  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( Kt  ran  F )  e.  Comp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843   ran crn 4706    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648    Cn ccn 16970   Compccmp 17129
This theorem is referenced by:  imacmp  17140  kgencn2  17268  bndth  18472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cn 16973  df-cmp 17130
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