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Theorem iskgen3 19120
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 19119 . 2  |-  ( J  e.  ran 𝑘Gen  <->  ( J  e. 
Top  /\  (𝑘Gen `  J
)  C_  J )
)
2 iskgen3.1 . . . . . . . . . 10  |-  X  = 
U. J
32toptopon 18536 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4 elkgen 19107 . . . . . . . . 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 2973 . . . . . . . . . 10  |-  x  e. 
_V
76elpw 3864 . . . . . . . . 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 1679 . . . 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 3343 . . . 4  |-  ( (𝑘Gen `  J )  C_  J  <->  A. x ( x  e.  (𝑘Gen `  J )  ->  x  e.  J )
)
15 df-ral 2718 . . . 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 1367    = wceq 1369    e. wcel 1756   A.wral 2713    i^i cin 3325    C_ wss 3326   ~Pcpw 3858   U.cuni 4089   ran crn 4839   ` cfv 5416  (class class class)co 6089   ↾t crest 14357   Topctop 18496  TopOnctopon 18497   Compccmp 18987  𝑘Genckgen 19104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  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 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-oadd 6922  df-er 7099  df-en 7309  df-fin 7312  df-fi 7659  df-rest 14359  df-topgen 14380  df-top 18501  df-bases 18503  df-topon 18504  df-cmp 18988  df-kgen 19105
This theorem is referenced by: (None)
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