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Theorem iskgen3 19923
Description: Derive the usual definition of "compactly generated". A topology is compactly generated if every subset of  X that is open in every compact subset is open. (Contributed by Mario Carneiro, 20-Mar-2015.)
Hypothesis
Ref Expression
iskgen3.1  |-  X  = 
U. J
Assertion
Ref Expression
iskgen3  |-  ( J  e.  ran 𝑘Gen  <->  ( J  e. 
Top  /\  A. x  e.  ~P  X ( A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  ->  x  e.  J
) ) )
Distinct variable groups:    x, k, J    k, X
Allowed substitution hint:    X( x)

Proof of Theorem iskgen3
StepHypRef Expression
1 iskgen2 19922 . 2  |-  ( J  e.  ran 𝑘Gen  <->  ( J  e. 
Top  /\  (𝑘Gen `  J
)  C_  J )
)
2 iskgen3.1 . . . . . . . . . 10  |-  X  = 
U. J
32toptopon 19307 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4 elkgen 19910 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  (𝑘Gen `  J )  <->  ( x  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) ) ) ) )
53, 4sylbi 195 . . . . . . . 8  |-  ( J  e.  Top  ->  (
x  e.  (𝑘Gen `  J
)  <->  ( x  C_  X  /\  A. k  e. 
~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) ) ) ) )
6 vex 3098 . . . . . . . . . 10  |-  x  e. 
_V
76elpw 4003 . . . . . . . . 9  |-  ( x  e.  ~P X  <->  x  C_  X
)
87anbi1i 695 . . . . . . . 8  |-  ( ( x  e.  ~P X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) )  <->  ( x  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) ) ) )
95, 8syl6bbr 263 . . . . . . 7  |-  ( J  e.  Top  ->  (
x  e.  (𝑘Gen `  J
)  <->  ( x  e. 
~P X  /\  A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) ) ) )
109imbi1d 317 . . . . . 6  |-  ( J  e.  Top  ->  (
( x  e.  (𝑘Gen `  J )  ->  x  e.  J )  <->  ( (
x  e.  ~P X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) )  ->  x  e.  J ) ) )
11 impexp 446 . . . . . 6  |-  ( ( ( x  e.  ~P X  /\  A. k  e. 
~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) ) )  ->  x  e.  J )  <->  ( x  e.  ~P X  ->  ( A. k  e. 
~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) )  ->  x  e.  J ) ) )
1210, 11syl6bb 261 . . . . 5  |-  ( J  e.  Top  ->  (
( x  e.  (𝑘Gen `  J )  ->  x  e.  J )  <->  ( x  e.  ~P X  ->  ( A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  ->  x  e.  J ) ) ) )
1312albidv 1700 . . . 4  |-  ( J  e.  Top  ->  ( A. x ( x  e.  (𝑘Gen `  J )  ->  x  e.  J )  <->  A. x ( x  e. 
~P X  ->  ( A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  ->  x  e.  J ) ) ) )
14 dfss2 3478 . . . 4  |-  ( (𝑘Gen `  J )  C_  J  <->  A. x ( x  e.  (𝑘Gen `  J )  ->  x  e.  J )
)
15 df-ral 2798 . . . 4  |-  ( A. x  e.  ~P  X
( A. k  e. 
~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) )  ->  x  e.  J )  <->  A. x
( x  e.  ~P X  ->  ( A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) )  ->  x  e.  J ) ) )
1613, 14, 153bitr4g 288 . . 3  |-  ( J  e.  Top  ->  (
(𝑘Gen `  J )  C_  J 
<-> 
A. x  e.  ~P  X ( A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) )  ->  x  e.  J ) ) )
1716pm5.32i 637 . 2  |-  ( ( J  e.  Top  /\  (𝑘Gen
`  J )  C_  J )  <->  ( J  e.  Top  /\  A. x  e.  ~P  X ( A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  ->  x  e.  J
) ) )
181, 17bitri 249 1  |-  ( J  e.  ran 𝑘Gen  <->  ( J  e. 
Top  /\  A. x  e.  ~P  X ( A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  ->  x  e.  J
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1381    = wceq 1383    e. wcel 1804   A.wral 2793    i^i cin 3460    C_ wss 3461   ~Pcpw 3997   U.cuni 4234   ran crn 4990   ` cfv 5578  (class class class)co 6281   ↾t crest 14695   Topctop 19267  TopOnctopon 19268   Compccmp 19759  𝑘Genckgen 19907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-oadd 7136  df-er 7313  df-en 7519  df-fin 7522  df-fi 7873  df-rest 14697  df-topgen 14718  df-top 19272  df-bases 19274  df-topon 19275  df-cmp 19760  df-kgen 19908
This theorem is referenced by: (None)
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