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Theorem kgencmp 19245
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 19240 . . . 4  |-  ( J  e.  Top  ->  (𝑘Gen `  J )  e.  Top )
21adantr 465 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  (𝑘Gen `  J )  e. 
Top )
3 kgenss 19243 . . . 4  |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
43adantr 465 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  J  C_  (𝑘Gen `  J ) )
5 ssrest 18907 . . 3  |-  ( ( (𝑘Gen `  J )  e. 
Top  /\  J  C_  (𝑘Gen `  J ) )  -> 
( Jt  K )  C_  (
(𝑘Gen `  J )t  K ) )
62, 4, 5syl2anc 661 . 2  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K ) 
C_  ( (𝑘Gen `  J
)t 
K ) )
7 cmptop 19125 . . . . . 6  |-  ( ( Jt  K )  e.  Comp  -> 
( Jt  K )  e.  Top )
87adantl 466 . . . . 5  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K )  e.  Top )
9 restrcl 18888 . . . . . 6  |-  ( ( Jt  K )  e.  Top  ->  ( J  e.  _V  /\  K  e.  _V )
)
109simprd 463 . . . . 5  |-  ( ( Jt  K )  e.  Top  ->  K  e.  _V )
118, 10syl 16 . . . 4  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  K  e.  _V )
12 restval 14479 . . . 4  |-  ( ( (𝑘Gen `  J )  e. 
Top  /\  K  e.  _V )  ->  ( (𝑘Gen `  J )t  K )  =  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) )
132, 11, 12syl2anc 661 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( (𝑘Gen `  J
)t 
K )  =  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) )
14 simpr 461 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  ->  x  e.  (𝑘Gen `  J
) )
15 simplr 754 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  -> 
( Jt  K )  e.  Comp )
16 kgeni 19237 . . . . . 6  |-  ( ( x  e.  (𝑘Gen `  J
)  /\  ( Jt  K
)  e.  Comp )  ->  ( x  i^i  K
)  e.  ( Jt  K ) )
1714, 15, 16syl2anc 661 . . . . 5  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  -> 
( x  i^i  K
)  e.  ( Jt  K ) )
18 eqid 2452 . . . . 5  |-  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )  =  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )
1917, 18fmptd 5971 . . . 4  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) : (𝑘Gen `  J
) --> ( Jt  K ) )
20 frn 5668 . . . 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 3493 . 2  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( (𝑘Gen `  J
)t 
K )  C_  ( Jt  K ) )
236, 22eqssd 3476 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 1370    e. wcel 1758   _Vcvv 3072    i^i cin 3430    C_ wss 3431    |-> cmpt 4453   ran crn 4944   -->wf 5517   ` cfv 5521  (class class class)co 6195   ↾t crest 14473   Topctop 18625   Compccmp 19116  𝑘Genckgen 19233
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-oadd 7029  df-er 7206  df-en 7416  df-fin 7419  df-fi 7767  df-rest 14475  df-topgen 14496  df-top 18630  df-bases 18632  df-topon 18633  df-cmp 19117  df-kgen 19234
This theorem is referenced by:  kgencmp2  19246  kgenidm  19247
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