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Theorem kgen2ss 17250
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 955 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  J  e.  (TopOn `  X ) )
2 elpwi 3633 . . . . . . . . 9  |-  ( k  e.  ~P X  -> 
k  C_  X )
3 resttopon 16892 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  k  C_  X )  ->  ( Jt  k )  e.  (TopOn `  k ) )
41, 2, 3syl2an 463 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Jt  k )  e.  (TopOn `  k ) )
5 simp2 956 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  K  e.  (TopOn `  X ) )
6 resttopon 16892 . . . . . . . . . . 11  |-  ( ( K  e.  (TopOn `  X )  /\  k  C_  X )  ->  ( Kt  k )  e.  (TopOn `  k ) )
75, 2, 6syl2an 463 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Kt  k )  e.  (TopOn `  k ) )
8 toponuni 16665 . . . . . . . . . 10  |-  ( ( Kt  k )  e.  (TopOn `  k )  ->  k  =  U. ( Kt  k ) )
97, 8syl 15 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  k  =  U. ( Kt  k ) )
109fveq2d 5529 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  (TopOn `  k )  =  (TopOn `  U. ( Kt  k ) ) )
114, 10eleqtrd 2359 . . . . . . 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 959 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  K  e.  (TopOn `  X )
)
13 topontop 16664 . . . . . . . . 9  |-  ( K  e.  (TopOn `  X
)  ->  K  e.  Top )
1412, 13syl 15 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  K  e.  Top )
15 simpl3 960 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  J  C_  K )
16 ssrest 16907 . . . . . . . 8  |-  ( ( K  e.  Top  /\  J  C_  K )  -> 
( Jt  k )  C_  ( Kt  k ) )
1714, 15, 16syl2anc 642 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Jt  k )  C_  ( Kt  k ) )
18 eqid 2283 . . . . . . . . . 10  |-  U. ( Kt  k )  =  U. ( Kt  k )
1918sscmp 17132 . . . . . . . . 9  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Kt  k )  e.  Comp  /\  ( Jt  k )  C_  ( Kt  k ) )  ->  ( Jt  k )  e.  Comp )
20193com23 1157 . . . . . . . 8  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Jt  k )  C_  ( Kt  k )  /\  ( Kt  k )  e.  Comp )  ->  ( Jt  k )  e.  Comp )
21203expia 1153 . . . . . . 7  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Jt  k )  C_  ( Kt  k ) )  -> 
( ( Kt  k )  e.  Comp  ->  ( Jt  k )  e.  Comp )
)
2211, 17, 21syl2anc 642 . . . . . 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 3179 . . . . . 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 68 . . . . 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 2621 . . . 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 548 . . 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 17231 . . . 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 976 . . 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 17231 . . . 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 977 . . 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 259 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( x  e.  (𝑘Gen `  J )  ->  x  e.  (𝑘Gen `  K
) ) )
3231ssrdv 3185 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   ` 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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