MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iskgen3 Structured version   Unicode version

Theorem iskgen3 19801
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 19800 . 2  |-  ( J  e.  ran 𝑘Gen  <->  ( J  e. 
Top  /\  (𝑘Gen `  J
)  C_  J )
)
2 iskgen3.1 . . . . . . . . . 10  |-  X  = 
U. J
32toptopon 19217 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4 elkgen 19788 . . . . . . . . 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 3116 . . . . . . . . . 10  |-  x  e. 
_V
76elpw 4016 . . . . . . . . 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 1689 . . . 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 3493 . . . 4  |-  ( (𝑘Gen `  J )  C_  J  <->  A. x ( x  e.  (𝑘Gen `  J )  ->  x  e.  J )
)
15 df-ral 2819 . . . 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 1377    = wceq 1379    e. wcel 1767   A.wral 2814    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   ran crn 5000   ` cfv 5587  (class class class)co 6283   ↾t crest 14675   Topctop 19177  TopOnctopon 19178   Compccmp 19668  𝑘Genckgen 19785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-oadd 7134  df-er 7311  df-en 7517  df-fin 7520  df-fi 7870  df-rest 14677  df-topgen 14698  df-top 19182  df-bases 19184  df-topon 19185  df-cmp 19669  df-kgen 19786
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator