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Theorem kgencmp 19077
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 19072 . . . 4  |-  ( J  e.  Top  ->  (𝑘Gen `  J )  e.  Top )
21adantr 462 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  (𝑘Gen `  J )  e. 
Top )
3 kgenss 19075 . . . 4  |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
43adantr 462 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  J  C_  (𝑘Gen `  J ) )
5 ssrest 18739 . . 3  |-  ( ( (𝑘Gen `  J )  e. 
Top  /\  J  C_  (𝑘Gen `  J ) )  -> 
( Jt  K )  C_  (
(𝑘Gen `  J )t  K ) )
62, 4, 5syl2anc 656 . 2  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K ) 
C_  ( (𝑘Gen `  J
)t 
K ) )
7 cmptop 18957 . . . . . 6  |-  ( ( Jt  K )  e.  Comp  -> 
( Jt  K )  e.  Top )
87adantl 463 . . . . 5  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K )  e.  Top )
9 restrcl 18720 . . . . . 6  |-  ( ( Jt  K )  e.  Top  ->  ( J  e.  _V  /\  K  e.  _V )
)
109simprd 460 . . . . 5  |-  ( ( Jt  K )  e.  Top  ->  K  e.  _V )
118, 10syl 16 . . . 4  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  K  e.  _V )
12 restval 14361 . . . 4  |-  ( ( (𝑘Gen `  J )  e. 
Top  /\  K  e.  _V )  ->  ( (𝑘Gen `  J )t  K )  =  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) )
132, 11, 12syl2anc 656 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( (𝑘Gen `  J
)t 
K )  =  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) )
14 simpr 458 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  ->  x  e.  (𝑘Gen `  J
) )
15 simplr 749 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  -> 
( Jt  K )  e.  Comp )
16 kgeni 19069 . . . . . 6  |-  ( ( x  e.  (𝑘Gen `  J
)  /\  ( Jt  K
)  e.  Comp )  ->  ( x  i^i  K
)  e.  ( Jt  K ) )
1714, 15, 16syl2anc 656 . . . . 5  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  -> 
( x  i^i  K
)  e.  ( Jt  K ) )
18 eqid 2441 . . . . 5  |-  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )  =  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )
1917, 18fmptd 5864 . . . 4  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) : (𝑘Gen `  J
) --> ( Jt  K ) )
20 frn 5562 . . . 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 16 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )  C_  ( Jt  K
) )
2213, 21eqsstrd 3387 . 2  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( (𝑘Gen `  J
)t 
K )  C_  ( Jt  K ) )
236, 22eqssd 3370 1  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K )  =  ( (𝑘Gen `  J
)t 
K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970    i^i cin 3324    C_ wss 3325    e. cmpt 4347   ran crn 4837   -->wf 5411   ` cfv 5415  (class class class)co 6090   ↾t crest 14355   Topctop 18457   Compccmp 18948  𝑘Genckgen 19065
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-oadd 6920  df-er 7097  df-en 7307  df-fin 7310  df-fi 7657  df-rest 14357  df-topgen 14378  df-top 18462  df-bases 18464  df-topon 18465  df-cmp 18949  df-kgen 19066
This theorem is referenced by:  kgencmp2  19078  kgenidm  19079
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