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Theorem kgen2ss 17544
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 957 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  J  e.  (TopOn `  X ) )
2 elpwi 3771 . . . . . . . . 9  |-  ( k  e.  ~P X  -> 
k  C_  X )
3 resttopon 17183 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  k  C_  X )  ->  ( Jt  k )  e.  (TopOn `  k ) )
41, 2, 3syl2an 464 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Jt  k )  e.  (TopOn `  k ) )
5 simp2 958 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  K  e.  (TopOn `  X ) )
6 resttopon 17183 . . . . . . . . . . 11  |-  ( ( K  e.  (TopOn `  X )  /\  k  C_  X )  ->  ( Kt  k )  e.  (TopOn `  k ) )
75, 2, 6syl2an 464 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Kt  k )  e.  (TopOn `  k ) )
8 toponuni 16951 . . . . . . . . . 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 5695 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  (TopOn `  k )  =  (TopOn `  U. ( Kt  k ) ) )
114, 10eleqtrd 2484 . . . . . . 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 961 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  K  e.  (TopOn `  X )
)
13 topontop 16950 . . . . . . . . 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 962 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  J  C_  K )
16 ssrest 17198 . . . . . . . 8  |-  ( ( K  e.  Top  /\  J  C_  K )  -> 
( Jt  k )  C_  ( Kt  k ) )
1714, 15, 16syl2anc 643 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Jt  k )  C_  ( Kt  k ) )
18 eqid 2408 . . . . . . . . . 10  |-  U. ( Kt  k )  =  U. ( Kt  k )
1918sscmp 17426 . . . . . . . . 9  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Kt  k )  e.  Comp  /\  ( Jt  k )  C_  ( Kt  k ) )  ->  ( Jt  k )  e.  Comp )
20193com23 1159 . . . . . . . 8  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Jt  k )  C_  ( Kt  k )  /\  ( Kt  k )  e.  Comp )  ->  ( Jt  k )  e.  Comp )
21203expia 1155 . . . . . . 7  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Jt  k )  C_  ( Kt  k ) )  -> 
( ( Kt  k )  e.  Comp  ->  ( Jt  k )  e.  Comp )
)
2211, 17, 21syl2anc 643 . . . . . 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 3311 . . . . . 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 70 . . . . 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 2748 . . . 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 549 . . 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 17525 . . . 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 978 . . 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 17525 . . . 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 979 . . 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 260 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( x  e.  (𝑘Gen `  J )  ->  x  e.  (𝑘Gen `  K
) ) )
3231ssrdv 3318 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670    i^i cin 3283    C_ wss 3284   ~Pcpw 3763   U.cuni 3979   ` cfv 5417  (class class class)co 6044   ↾t crest 13607   Topctop 16917  TopOnctopon 16918   Compccmp 17407  𝑘Genckgen 17522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-recs 6596  df-rdg 6631  df-oadd 6691  df-er 6868  df-en 7073  df-fin 7076  df-fi 7378  df-rest 13609  df-topgen 13626  df-top 16922  df-bases 16924  df-topon 16925  df-cmp 17408  df-kgen 17523
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