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Theorem iskgen3 17244
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 17243 . 2  |-  ( J  e.  ran 𝑘Gen  <->  ( J  e. 
Top  /\  (𝑘Gen `  J
)  C_  J )
)
2 iskgen3.1 . . . . . . . . . 10  |-  X  = 
U. J
32toptopon 16671 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4 elkgen 17231 . . . . . . . . 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 187 . . . . . . . 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 2791 . . . . . . . . . 10  |-  x  e. 
_V
76elpw 3631 . . . . . . . . 9  |-  ( x  e.  ~P X  <->  x  C_  X
)
87anbi1i 676 . . . . . . . 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 254 . . . . . . 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 308 . . . . . 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 433 . . . . . 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 252 . . . . 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 1611 . . . 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 3169 . . . 4  |-  ( (𝑘Gen `  J )  C_  J  <->  A. x ( x  e.  (𝑘Gen `  J )  ->  x  e.  J )
)
15 df-ral 2548 . . . 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 279 . . 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 618 . 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 240 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   ran crn 4690   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631  TopOnctopon 16632   Compccmp 17113  𝑘Genckgen 17228
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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