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Theorem kgen2ss 17356
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 3709 . . . . . . . . 9  |-  ( k  e.  ~P X  -> 
k  C_  X )
3 resttopon 16998 . . . . . . . . 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 16998 . . . . . . . . . . 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 16771 . . . . . . . . . 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 5612 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  /\  k  e.  ~P X )  ->  (TopOn `  k )  =  (TopOn `  U. ( Kt  k ) ) )
114, 10eleqtrd 2434 . . . . . . 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 16770 . . . . . . . . 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 17013 . . . . . . . 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 2358 . . . . . . . . . 10  |-  U. ( Kt  k )  =  U. ( Kt  k )
1918sscmp 17238 . . . . . . . . 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 3255 . . . . . 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 2697 . . . 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 17337 . . . 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 17337 . . . 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 3261 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 1642    e. wcel 1710   A.wral 2619    i^i cin 3227    C_ wss 3228   ~Pcpw 3701   U.cuni 3908   ` cfv 5337  (class class class)co 5945   ↾t crest 13424   Topctop 16737  TopOnctopon 16738   Compccmp 17219  𝑘Genckgen 17334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-recs 6475  df-rdg 6510  df-oadd 6570  df-er 6747  df-en 6952  df-fin 6955  df-fi 7255  df-rest 13426  df-topgen 13443  df-top 16742  df-bases 16744  df-topon 16745  df-cmp 17220  df-kgen 17335
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