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Theorem kgenidm 17258
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 17252 . . . 4  |- 𝑘Gen : Top --> Top
2 ffn 5405 . . . 4  |-  (𝑘Gen : Top --> Top 
-> 𝑘Gen 
Fn  Top )
3 fvelrnb 5586 . . . 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 2296 . . . . . . . . . . . 12  |-  U. j  =  U. j
65toptopon 16687 . . . . . . . . . . 11  |-  ( j  e.  Top  <->  j  e.  (TopOn `  U. j ) )
7 kgentopon 17249 . . . . . . . . . . 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 17249 . . . . . . . . . 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 16683 . . . . . . . . 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 17257 . . . . . . . . . . . . . 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 17248 . . . . . . . . . . . 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 17256 . . . . . . . . . . . 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 2373 . . . . . . . . . 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 2639 . . . . . . . 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 17247 . . . . . . . . 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 3198 . . . . 5  |-  ( j  e.  Top  ->  (𝑘Gen `  (𝑘Gen
`  j ) ) 
C_  (𝑘Gen `  j ) )
31 fveq2 5541 . . . . . 6  |-  ( (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  (𝑘Gen `  j ) )  =  (𝑘Gen `  J ) )
32 id 19 . . . . . 6  |-  ( (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  j )  =  J )
3331, 32sseq12d 3220 . . . . 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 2674 . . 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 17253 . . 3  |-  ( J  e.  ran 𝑘Gen  ->  J  e.  Top )
38 kgenss 17254 . . 3  |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
3937, 38syl 15 . 2  |-  ( J  e.  ran 𝑘Gen  ->  J  C_  (𝑘Gen `  J ) )
4036, 39eqssd 3209 1  |-  ( J  e.  ran 𝑘Gen  ->  (𝑘Gen `  J
)  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648   Compccmp 17129  𝑘Genckgen 17244
This theorem is referenced by:  iskgen2  17259  kgencn3  17269  txkgen  17362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cmp 17130  df-kgen 17245
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