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Theorem kgenidm 17532
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 17526 . . . 4  |- 𝑘Gen : Top --> Top
2 ffn 5550 . . . 4  |-  (𝑘Gen : Top --> Top 
-> 𝑘Gen 
Fn  Top )
3 fvelrnb 5733 . . . 4  |-  (𝑘Gen  Fn  Top  ->  ( J  e.  ran 𝑘Gen  <->  E. j  e.  Top  (𝑘Gen `  j )  =  J ) )
41, 2, 3mp2b 10 . . 3  |-  ( J  e.  ran 𝑘Gen  <->  E. j  e.  Top  (𝑘Gen
`  j )  =  J )
5 eqid 2404 . . . . . . . . . . . 12  |-  U. j  =  U. j
65toptopon 16953 . . . . . . . . . . 11  |-  ( j  e.  Top  <->  j  e.  (TopOn `  U. j ) )
7 kgentopon 17523 . . . . . . . . . . 11  |-  ( j  e.  (TopOn `  U. j )  ->  (𝑘Gen `  j )  e.  (TopOn `  U. j ) )
86, 7sylbi 188 . . . . . . . . . 10  |-  ( j  e.  Top  ->  (𝑘Gen `  j )  e.  (TopOn `  U. j ) )
9 kgentopon 17523 . . . . . . . . . 10  |-  ( (𝑘Gen `  j )  e.  (TopOn `  U. j )  -> 
(𝑘Gen `  (𝑘Gen `  j ) )  e.  (TopOn `  U. j ) )
108, 9syl 16 . . . . . . . . 9  |-  ( j  e.  Top  ->  (𝑘Gen `  (𝑘Gen
`  j ) )  e.  (TopOn `  U. j ) )
11 toponss 16949 . . . . . . . . 9  |-  ( ( (𝑘Gen `  (𝑘Gen `  j ) )  e.  (TopOn `  U. j )  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  x  C_  U. j
)
1210, 11sylan 458 . . . . . . . 8  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  x  C_  U. j
)
13 simplr 732 . . . . . . . . . . . 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 17531 . . . . . . . . . . . . . 14  |-  ( j  e.  Top  ->  (
( jt  k )  e. 
Comp 
<->  ( (𝑘Gen `  j )t  k )  e.  Comp ) )
1514biimpa 471 . . . . . . . . . . . . 13  |-  ( ( j  e.  Top  /\  ( jt  k )  e. 
Comp )  ->  (
(𝑘Gen `  j )t  k )  e.  Comp )
1615ad2ant2rl 730 . . . . . . . . . . . 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 17522 . . . . . . . . . . . 12  |-  ( ( x  e.  (𝑘Gen `  (𝑘Gen `  j ) )  /\  ( (𝑘Gen `  j )t  k )  e.  Comp )  ->  (
x  i^i  k )  e.  ( (𝑘Gen `  j )t  k ) )
1813, 16, 17syl2anc 643 . . . . . . . . . . 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 17530 . . . . . . . . . . . 12  |-  ( ( j  e.  Top  /\  ( jt  k )  e. 
Comp )  ->  (
jt  k )  =  ( (𝑘Gen `  j )t  k ) )
2019ad2ant2rl 730 . . . . . . . . . . 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 2481 . . . . . . . . . 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 599 . . . . . . . . 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 2749 . . . . . . . 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 444 . . . . . . . . . 10  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  j  e.  Top )
2524, 6sylib 189 . . . . . . . . 9  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  j  e.  (TopOn `  U. j ) )
26 elkgen 17521 . . . . . . . . 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 16 . . . . . . . 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 889 . . . . . . 7  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  x  e.  (𝑘Gen `  j ) )
2928ex 424 . . . . . 6  |-  ( j  e.  Top  ->  (
x  e.  (𝑘Gen `  (𝑘Gen `  j ) )  ->  x  e.  (𝑘Gen `  j
) ) )
3029ssrdv 3314 . . . . 5  |-  ( j  e.  Top  ->  (𝑘Gen `  (𝑘Gen
`  j ) ) 
C_  (𝑘Gen `  j ) )
31 fveq2 5687 . . . . . 6  |-  ( (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  (𝑘Gen `  j ) )  =  (𝑘Gen `  J ) )
32 id 20 . . . . . 6  |-  ( (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  j )  =  J )
3331, 32sseq12d 3337 . . . . 5  |-  ( (𝑘Gen `  j )  =  J  ->  ( (𝑘Gen `  (𝑘Gen `  j ) )  C_  (𝑘Gen
`  j )  <->  (𝑘Gen `  J
)  C_  J )
)
3430, 33syl5ibcom 212 . . . 4  |-  ( j  e.  Top  ->  (
(𝑘Gen `  j )  =  J  ->  (𝑘Gen `  J
)  C_  J )
)
3534rexlimiv 2784 . . 3  |-  ( E. j  e.  Top  (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  J )  C_  J )
364, 35sylbi 188 . 2  |-  ( J  e.  ran 𝑘Gen  ->  (𝑘Gen `  J
)  C_  J )
37 kgentop 17527 . . 3  |-  ( J  e.  ran 𝑘Gen  ->  J  e.  Top )
38 kgenss 17528 . . 3  |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
3937, 38syl 16 . 2  |-  ( J  e.  ran 𝑘Gen  ->  J  C_  (𝑘Gen `  J ) )
4036, 39eqssd 3325 1  |-  ( J  e.  ran 𝑘Gen  ->  (𝑘Gen `  J
)  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   ran crn 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   ↾t crest 13603   Topctop 16913  TopOnctopon 16914   Compccmp 17403  𝑘Genckgen 17518
This theorem is referenced by:  iskgen2  17533  kgencn3  17543  txkgen  17637
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-oadd 6687  df-er 6864  df-en 7069  df-fin 7072  df-fi 7374  df-rest 13605  df-topgen 13622  df-top 16918  df-bases 16920  df-topon 16921  df-cmp 17404  df-kgen 17519
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