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Theorem kgenidm 17242
Description: The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgenidm  |-  ( J  e.  ran 𝑘Gen  ->  (𝑘Gen `  J
)  =  J )

Proof of Theorem kgenidm
Dummy variables  j 
k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kgenf 17236 . . . 4  |- 𝑘Gen : Top --> Top
2 ffn 5389 . . . 4  |-  (𝑘Gen : Top --> Top 
-> 𝑘Gen 
Fn  Top )
3 fvelrnb 5570 . . . 4  |-  (𝑘Gen  Fn  Top  ->  ( J  e.  ran 𝑘Gen  <->  E. j  e.  Top  (𝑘Gen `  j )  =  J ) )
41, 2, 3mp2b 9 . . 3  |-  ( J  e.  ran 𝑘Gen  <->  E. j  e.  Top  (𝑘Gen
`  j )  =  J )
5 eqid 2283 . . . . . . . . . . . 12  |-  U. j  =  U. j
65toptopon 16671 . . . . . . . . . . 11  |-  ( j  e.  Top  <->  j  e.  (TopOn `  U. j ) )
7 kgentopon 17233 . . . . . . . . . . 11  |-  ( j  e.  (TopOn `  U. j )  ->  (𝑘Gen `  j )  e.  (TopOn `  U. j ) )
86, 7sylbi 187 . . . . . . . . . 10  |-  ( j  e.  Top  ->  (𝑘Gen `  j )  e.  (TopOn `  U. j ) )
9 kgentopon 17233 . . . . . . . . . 10  |-  ( (𝑘Gen `  j )  e.  (TopOn `  U. j )  -> 
(𝑘Gen `  (𝑘Gen `  j ) )  e.  (TopOn `  U. j ) )
108, 9syl 15 . . . . . . . . 9  |-  ( j  e.  Top  ->  (𝑘Gen `  (𝑘Gen
`  j ) )  e.  (TopOn `  U. j ) )
11 toponss 16667 . . . . . . . . 9  |-  ( ( (𝑘Gen `  (𝑘Gen `  j ) )  e.  (TopOn `  U. j )  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  x  C_  U. j
)
1210, 11sylan 457 . . . . . . . 8  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  x  C_  U. j
)
13 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  /\  ( k  e. 
~P U. j  /\  (
jt  k )  e.  Comp ) )  ->  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )
14 kgencmp2 17241 . . . . . . . . . . . . . 14  |-  ( j  e.  Top  ->  (
( jt  k )  e. 
Comp 
<->  ( (𝑘Gen `  j )t  k )  e.  Comp ) )
1514biimpa 470 . . . . . . . . . . . . 13  |-  ( ( j  e.  Top  /\  ( jt  k )  e. 
Comp )  ->  (
(𝑘Gen `  j )t  k )  e.  Comp )
1615ad2ant2rl 729 . . . . . . . . . . . 12  |-  ( ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  /\  ( k  e. 
~P U. j  /\  (
jt  k )  e.  Comp ) )  ->  (
(𝑘Gen `  j )t  k )  e.  Comp )
17 kgeni 17232 . . . . . . . . . . . 12  |-  ( ( x  e.  (𝑘Gen `  (𝑘Gen `  j ) )  /\  ( (𝑘Gen `  j )t  k )  e.  Comp )  ->  (
x  i^i  k )  e.  ( (𝑘Gen `  j )t  k ) )
1813, 16, 17syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  /\  ( k  e. 
~P U. j  /\  (
jt  k )  e.  Comp ) )  ->  (
x  i^i  k )  e.  ( (𝑘Gen `  j )t  k ) )
19 kgencmp 17240 . . . . . . . . . . . 12  |-  ( ( j  e.  Top  /\  ( jt  k )  e. 
Comp )  ->  (
jt  k )  =  ( (𝑘Gen `  j )t  k ) )
2019ad2ant2rl 729 . . . . . . . . . . 11  |-  ( ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  /\  ( k  e. 
~P U. j  /\  (
jt  k )  e.  Comp ) )  ->  (
jt  k )  =  ( (𝑘Gen `  j )t  k ) )
2118, 20eleqtrrd 2360 . . . . . . . . . 10  |-  ( ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  /\  ( k  e. 
~P U. j  /\  (
jt  k )  e.  Comp ) )  ->  (
x  i^i  k )  e.  ( jt  k ) )
2221expr 598 . . . . . . . . 9  |-  ( ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  /\  k  e.  ~P U. j )  ->  (
( jt  k )  e. 
Comp  ->  ( x  i^i  k )  e.  ( jt  k ) ) )
2322ralrimiva 2626 . . . . . . . 8  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  A. k  e.  ~P  U. j ( ( jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( jt  k ) ) )
24 simpl 443 . . . . . . . . . 10  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  j  e.  Top )
2524, 6sylib 188 . . . . . . . . 9  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  j  e.  (TopOn `  U. j ) )
26 elkgen 17231 . . . . . . . . 9  |-  ( j  e.  (TopOn `  U. j )  ->  (
x  e.  (𝑘Gen `  j
)  <->  ( x  C_  U. j  /\  A. k  e.  ~P  U. j ( ( jt  k )  e. 
Comp  ->  ( x  i^i  k )  e.  ( jt  k ) ) ) ) )
2725, 26syl 15 . . . . . . . 8  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  ( x  e.  (𝑘Gen `  j )  <->  ( x  C_ 
U. j  /\  A. k  e.  ~P  U. j
( ( jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( jt  k ) ) ) ) )
2812, 23, 27mpbir2and 888 . . . . . . 7  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  x  e.  (𝑘Gen `  j ) )
2928ex 423 . . . . . 6  |-  ( j  e.  Top  ->  (
x  e.  (𝑘Gen `  (𝑘Gen `  j ) )  ->  x  e.  (𝑘Gen `  j
) ) )
3029ssrdv 3185 . . . . 5  |-  ( j  e.  Top  ->  (𝑘Gen `  (𝑘Gen
`  j ) ) 
C_  (𝑘Gen `  j ) )
31 fveq2 5525 . . . . . 6  |-  ( (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  (𝑘Gen `  j ) )  =  (𝑘Gen `  J ) )
32 id 19 . . . . . 6  |-  ( (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  j )  =  J )
3331, 32sseq12d 3207 . . . . 5  |-  ( (𝑘Gen `  j )  =  J  ->  ( (𝑘Gen `  (𝑘Gen `  j ) )  C_  (𝑘Gen
`  j )  <->  (𝑘Gen `  J
)  C_  J )
)
3430, 33syl5ibcom 211 . . . 4  |-  ( j  e.  Top  ->  (
(𝑘Gen `  j )  =  J  ->  (𝑘Gen `  J
)  C_  J )
)
3534rexlimiv 2661 . . 3  |-  ( E. j  e.  Top  (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  J )  C_  J )
364, 35sylbi 187 . 2  |-  ( J  e.  ran 𝑘Gen  ->  (𝑘Gen `  J
)  C_  J )
37 kgentop 17237 . . 3  |-  ( J  e.  ran 𝑘Gen  ->  J  e.  Top )
38 kgenss 17238 . . 3  |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
3937, 38syl 15 . 2  |-  ( J  e.  ran 𝑘Gen  ->  J  C_  (𝑘Gen `  J ) )
4036, 39eqssd 3196 1  |-  ( J  e.  ran 𝑘Gen  ->  (𝑘Gen `  J
)  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631  TopOnctopon 16632   Compccmp 17113  𝑘Genckgen 17228
This theorem is referenced by:  iskgen2  17243  kgencn3  17253  txkgen  17346
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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