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Theorem kgen2ss 19784
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 991 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  J  e.  (TopOn `  X ) )
2 elpwi 4012 . . . . . . . . 9  |-  ( k  e.  ~P X  -> 
k  C_  X )
3 resttopon 19421 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  k  C_  X )  ->  ( Jt  k )  e.  (TopOn `  k ) )
41, 2, 3syl2an 477 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Jt  k )  e.  (TopOn `  k ) )
5 simp2 992 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  K  e.  (TopOn `  X ) )
6 resttopon 19421 . . . . . . . . . . 11  |-  ( ( K  e.  (TopOn `  X )  /\  k  C_  X )  ->  ( Kt  k )  e.  (TopOn `  k ) )
75, 2, 6syl2an 477 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Kt  k )  e.  (TopOn `  k ) )
8 toponuni 19188 . . . . . . . . . 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 5861 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  (TopOn `  k )  =  (TopOn `  U. ( Kt  k ) ) )
114, 10eleqtrd 2550 . . . . . . 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 995 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  K  e.  (TopOn `  X )
)
13 topontop 19187 . . . . . . . . 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 996 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  J  C_  K )
16 ssrest 19436 . . . . . . . 8  |-  ( ( K  e.  Top  /\  J  C_  K )  -> 
( Jt  k )  C_  ( Kt  k ) )
1714, 15, 16syl2anc 661 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  ( Jt  k )  C_  ( Kt  k ) )
18 eqid 2460 . . . . . . . . . 10  |-  U. ( Kt  k )  =  U. ( Kt  k )
1918sscmp 19664 . . . . . . . . 9  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Kt  k )  e.  Comp  /\  ( Jt  k )  C_  ( Kt  k ) )  ->  ( Jt  k )  e.  Comp )
20193com23 1197 . . . . . . . 8  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Jt  k )  C_  ( Kt  k )  /\  ( Kt  k )  e.  Comp )  ->  ( Jt  k )  e.  Comp )
21203expia 1193 . . . . . . 7  |-  ( ( ( Jt  k )  e.  (TopOn `  U. ( Kt  k ) )  /\  ( Jt  k )  C_  ( Kt  k ) )  -> 
( ( Kt  k )  e.  Comp  ->  ( Jt  k )  e.  Comp )
)
2211, 17, 21syl2anc 661 . . . . . 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 3496 . . . . . 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 74 . . . . 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 2865 . . . 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 565 . . 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 19765 . . . 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 1012 . . 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 19765 . . . 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 1013 . . 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 268 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( x  e.  (𝑘Gen `  J )  ->  x  e.  (𝑘Gen `  K
) ) )
3231ssrdv 3503 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  (𝑘Gen `  J )  C_  (𝑘Gen
`  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807    i^i cin 3468    C_ wss 3469   ~Pcpw 4003   U.cuni 4238   ` cfv 5579  (class class class)co 6275   ↾t crest 14665   Topctop 19154  TopOnctopon 19155   Compccmp 19645  𝑘Genckgen 19762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-oadd 7124  df-er 7301  df-en 7507  df-fin 7510  df-fi 7860  df-rest 14667  df-topgen 14688  df-top 19159  df-bases 19161  df-topon 19162  df-cmp 19646  df-kgen 19763
This theorem is referenced by: (None)
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