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Theorem kgencmp 17240
Description: The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgencmp  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K )  =  ( (𝑘Gen `  J
)t 
K ) )

Proof of Theorem kgencmp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 kgenftop 17235 . . . 4  |-  ( J  e.  Top  ->  (𝑘Gen `  J )  e.  Top )
21adantr 451 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  (𝑘Gen `  J )  e. 
Top )
3 kgenss 17238 . . . 4  |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
43adantr 451 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  J  C_  (𝑘Gen `  J ) )
5 ssrest 16907 . . 3  |-  ( ( (𝑘Gen `  J )  e. 
Top  /\  J  C_  (𝑘Gen `  J ) )  -> 
( Jt  K )  C_  (
(𝑘Gen `  J )t  K ) )
62, 4, 5syl2anc 642 . 2  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K ) 
C_  ( (𝑘Gen `  J
)t 
K ) )
7 cmptop 17122 . . . . . 6  |-  ( ( Jt  K )  e.  Comp  -> 
( Jt  K )  e.  Top )
87adantl 452 . . . . 5  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K )  e.  Top )
9 restrcl 16888 . . . . . 6  |-  ( ( Jt  K )  e.  Top  ->  ( J  e.  _V  /\  K  e.  _V )
)
109simprd 449 . . . . 5  |-  ( ( Jt  K )  e.  Top  ->  K  e.  _V )
118, 10syl 15 . . . 4  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  K  e.  _V )
12 restval 13331 . . . 4  |-  ( ( (𝑘Gen `  J )  e. 
Top  /\  K  e.  _V )  ->  ( (𝑘Gen `  J )t  K )  =  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) )
132, 11, 12syl2anc 642 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( (𝑘Gen `  J
)t 
K )  =  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) )
14 simpr 447 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  ->  x  e.  (𝑘Gen `  J
) )
15 simplr 731 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  -> 
( Jt  K )  e.  Comp )
16 kgeni 17232 . . . . . 6  |-  ( ( x  e.  (𝑘Gen `  J
)  /\  ( Jt  K
)  e.  Comp )  ->  ( x  i^i  K
)  e.  ( Jt  K ) )
1714, 15, 16syl2anc 642 . . . . 5  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  -> 
( x  i^i  K
)  e.  ( Jt  K ) )
18 eqid 2283 . . . . 5  |-  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )  =  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )
1917, 18fmptd 5684 . . . 4  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) : (𝑘Gen `  J
) --> ( Jt  K ) )
20 frn 5395 . . . 4  |-  ( ( x  e.  (𝑘Gen `  J
)  |->  ( x  i^i 
K ) ) : (𝑘Gen `  J ) --> ( Jt  K )  ->  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) 
C_  ( Jt  K ) )
2119, 20syl 15 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )  C_  ( Jt  K
) )
2213, 21eqsstrd 3212 . 2  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( (𝑘Gen `  J
)t 
K )  C_  ( Jt  K ) )
236, 22eqssd 3196 1  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K )  =  ( (𝑘Gen `  J
)t 
K ) )
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    e. cmpt 4077   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631   Compccmp 17113  𝑘Genckgen 17228
This theorem is referenced by:  kgencmp2  17241  kgenidm  17242
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-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cmp 17114  df-kgen 17229
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