MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kgen2ss Structured version   Unicode version

Theorem kgen2ss 17592
Description: The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
kgen2ss  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  (𝑘Gen `  J )  C_  (𝑘Gen
`  K ) )

Proof of Theorem kgen2ss
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 958 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  J  e.  (TopOn `  X ) )
2 elpwi 3809 . . . . . . . . 9  |-  ( k  e.  ~P X  -> 
k  C_  X )
3 resttopon 17230 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  k  C_  X )  ->  ( Jt  k )  e.  (TopOn `  k ) )
41, 2, 3syl2an 465 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Jt  k )  e.  (TopOn `  k ) )
5 simp2 959 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  K  e.  (TopOn `  X ) )
6 resttopon 17230 . . . . . . . . . . 11  |-  ( ( K  e.  (TopOn `  X )  /\  k  C_  X )  ->  ( Kt  k )  e.  (TopOn `  k ) )
75, 2, 6syl2an 465 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Kt  k )  e.  (TopOn `  k ) )
8 toponuni 16997 . . . . . . . . . 10  |-  ( ( Kt  k )  e.  (TopOn `  k )  ->  k  =  U. ( Kt  k ) )
97, 8syl 16 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  k  =  U. ( Kt  k ) )
109fveq2d 5735 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  (TopOn `  k )  =  (TopOn `  U. ( Kt  k ) ) )
114, 10eleqtrd 2514 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) ) )
12 simpl2 962 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  K  e.  (TopOn `  X )
)
13 topontop 16996 . . . . . . . . 9  |-  ( K  e.  (TopOn `  X
)  ->  K  e.  Top )
1412, 13syl 16 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  K  e.  Top )
15 simpl3 963 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  J  C_  K )
16 ssrest 17245 . . . . . . . 8  |-  ( ( K  e.  Top  /\  J  C_  K )  -> 
( Jt  k )  C_  ( Kt  k ) )
1714, 15, 16syl2anc 644 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Jt  k )  C_  ( Kt  k ) )
18 eqid 2438 . . . . . . . . . 10  |-  U. ( Kt  k )  =  U. ( Kt  k )
1918sscmp 17473 . . . . . . . . 9  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Kt  k )  e.  Comp  /\  ( Jt  k )  C_  ( Kt  k ) )  ->  ( Jt  k )  e.  Comp )
20193com23 1160 . . . . . . . 8  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Jt  k )  C_  ( Kt  k )  /\  ( Kt  k )  e.  Comp )  ->  ( Jt  k )  e.  Comp )
21203expia 1156 . . . . . . 7  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Jt  k )  C_  ( Kt  k ) )  -> 
( ( Kt  k )  e.  Comp  ->  ( Jt  k )  e.  Comp )
)
2211, 17, 21syl2anc 644 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  (
( Kt  k )  e. 
Comp  ->  ( Jt  k )  e.  Comp ) )
2317sseld 3349 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  (
( x  i^i  k
)  e.  ( Jt  k )  ->  ( x  i^i  k )  e.  ( Kt  k ) ) )
2422, 23imim12d 71 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  (
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  ->  ( ( Kt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Kt  k ) ) ) )
2524ralimdva 2786 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) )  ->  A. k  e.  ~P  X ( ( Kt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Kt  k ) ) ) )
2625anim2d 550 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( ( x 
C_  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 ( ( Kt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Kt  k ) ) ) ) )
27 elkgen 17573 . . . 4  |-  ( 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 ) ) ) ) )
28273ad2ant1 979 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( x  e.  (𝑘Gen `  J )  <->  ( x  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) ) ) ) )
29 elkgen 17573 . . . 4  |-  ( K  e.  (TopOn `  X
)  ->  ( x  e.  (𝑘Gen `  K )  <->  ( x  C_  X  /\  A. k  e.  ~P  X ( ( Kt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Kt  k ) ) ) ) )
30293ad2ant2 980 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( x  e.  (𝑘Gen `  K )  <->  ( x  C_  X  /\  A. k  e.  ~P  X ( ( Kt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Kt  k ) ) ) ) )
3126, 28, 303imtr4d 261 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( x  e.  (𝑘Gen `  J )  ->  x  e.  (𝑘Gen `  K
) ) )
3231ssrdv 3356 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  (𝑘Gen `  J )  C_  (𝑘Gen
`  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ` cfv 5457  (class class class)co 6084   ↾t crest 13653   Topctop 16963  TopOnctopon 16964   Compccmp 17454  𝑘Genckgen 17570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-oadd 6731  df-er 6908  df-en 7113  df-fin 7116  df-fi 7419  df-rest 13655  df-topgen 13672  df-top 16968  df-bases 16970  df-topon 16971  df-cmp 17455  df-kgen 17571
  Copyright terms: Public domain W3C validator