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Theorem iskgen3 17260
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 17259 . 2  |-  ( J  e.  ran 𝑘Gen  <->  ( J  e. 
Top  /\  (𝑘Gen `  J
)  C_  J )
)
2 iskgen3.1 . . . . . . . . . 10  |-  X  = 
U. J
32toptopon 16687 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4 elkgen 17247 . . . . . . . . 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 2804 . . . . . . . . . 10  |-  x  e. 
_V
76elpw 3644 . . . . . . . . 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 1615 . . . 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 3182 . . . 4  |-  ( (𝑘Gen `  J )  C_  J  <->  A. x ( x  e.  (𝑘Gen `  J )  ->  x  e.  J )
)
15 df-ral 2561 . . . 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 1530    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   ran crn 4706   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648   Compccmp 17129  𝑘Genckgen 17244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cmp 17130  df-kgen 17245
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