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Theorem kgenidm 17580
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 17574 . . . 4  |- 𝑘Gen : Top --> Top
2 ffn 5592 . . . 4  |-  (𝑘Gen : Top --> Top 
-> 𝑘Gen 
Fn  Top )
3 fvelrnb 5775 . . . 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 2437 . . . . . . . . . . . 12  |-  U. j  =  U. j
65toptopon 16999 . . . . . . . . . . 11  |-  ( j  e.  Top  <->  j  e.  (TopOn `  U. j ) )
7 kgentopon 17571 . . . . . . . . . . 11  |-  ( j  e.  (TopOn `  U. j )  ->  (𝑘Gen `  j )  e.  (TopOn `  U. j ) )
86, 7sylbi 189 . . . . . . . . . 10  |-  ( j  e.  Top  ->  (𝑘Gen `  j )  e.  (TopOn `  U. j ) )
9 kgentopon 17571 . . . . . . . . . 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 16995 . . . . . . . . 9  |-  ( ( (𝑘Gen `  (𝑘Gen `  j ) )  e.  (TopOn `  U. j )  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  x  C_  U. j
)
1210, 11sylan 459 . . . . . . . 8  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  x  C_  U. j
)
13 simplr 733 . . . . . . . . . . . 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 17579 . . . . . . . . . . . . . 14  |-  ( j  e.  Top  ->  (
( jt  k )  e. 
Comp 
<->  ( (𝑘Gen `  j )t  k )  e.  Comp ) )
1514biimpa 472 . . . . . . . . . . . . 13  |-  ( ( j  e.  Top  /\  ( jt  k )  e. 
Comp )  ->  (
(𝑘Gen `  j )t  k )  e.  Comp )
1615ad2ant2rl 731 . . . . . . . . . . . 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 17570 . . . . . . . . . . . 12  |-  ( ( x  e.  (𝑘Gen `  (𝑘Gen `  j ) )  /\  ( (𝑘Gen `  j )t  k )  e.  Comp )  ->  (
x  i^i  k )  e.  ( (𝑘Gen `  j )t  k ) )
1813, 16, 17syl2anc 644 . . . . . . . . . . 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 17578 . . . . . . . . . . . 12  |-  ( ( j  e.  Top  /\  ( jt  k )  e. 
Comp )  ->  (
jt  k )  =  ( (𝑘Gen `  j )t  k ) )
2019ad2ant2rl 731 . . . . . . . . . . 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 2514 . . . . . . . . . 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 600 . . . . . . . . 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 2790 . . . . . . . 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 445 . . . . . . . . . 10  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  j  e.  Top )
2524, 6sylib 190 . . . . . . . . 9  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  j  e.  (TopOn `  U. j ) )
26 elkgen 17569 . . . . . . . . 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 890 . . . . . . 7  |-  ( ( j  e.  Top  /\  x  e.  (𝑘Gen `  (𝑘Gen `  j ) ) )  ->  x  e.  (𝑘Gen `  j ) )
2928ex 425 . . . . . 6  |-  ( j  e.  Top  ->  (
x  e.  (𝑘Gen `  (𝑘Gen `  j ) )  ->  x  e.  (𝑘Gen `  j
) ) )
3029ssrdv 3355 . . . . 5  |-  ( j  e.  Top  ->  (𝑘Gen `  (𝑘Gen
`  j ) ) 
C_  (𝑘Gen `  j ) )
31 fveq2 5729 . . . . . 6  |-  ( (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  (𝑘Gen `  j ) )  =  (𝑘Gen `  J ) )
32 id 21 . . . . . 6  |-  ( (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  j )  =  J )
3331, 32sseq12d 3378 . . . . 5  |-  ( (𝑘Gen `  j )  =  J  ->  ( (𝑘Gen `  (𝑘Gen `  j ) )  C_  (𝑘Gen
`  j )  <->  (𝑘Gen `  J
)  C_  J )
)
3430, 33syl5ibcom 213 . . . 4  |-  ( j  e.  Top  ->  (
(𝑘Gen `  j )  =  J  ->  (𝑘Gen `  J
)  C_  J )
)
3534rexlimiv 2825 . . 3  |-  ( E. j  e.  Top  (𝑘Gen `  j )  =  J  ->  (𝑘Gen `  J )  C_  J )
364, 35sylbi 189 . 2  |-  ( J  e.  ran 𝑘Gen  ->  (𝑘Gen `  J
)  C_  J )
37 kgentop 17575 . . 3  |-  ( J  e.  ran 𝑘Gen  ->  J  e.  Top )
38 kgenss 17576 . . 3  |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
3937, 38syl 16 . 2  |-  ( J  e.  ran 𝑘Gen  ->  J  C_  (𝑘Gen `  J ) )
4036, 39eqssd 3366 1  |-  ( J  e.  ran 𝑘Gen  ->  (𝑘Gen `  J
)  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707    i^i cin 3320    C_ wss 3321   ~Pcpw 3800   U.cuni 4016   ran crn 4880    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082   ↾t crest 13649   Topctop 16959  TopOnctopon 16960   Compccmp 17450  𝑘Genckgen 17566
This theorem is referenced by:  iskgen2  17581  kgencn3  17591  txkgen  17685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-recs 6634  df-rdg 6669  df-oadd 6729  df-er 6906  df-en 7111  df-fin 7114  df-fi 7417  df-rest 13651  df-topgen 13668  df-top 16964  df-bases 16966  df-topon 16967  df-cmp 17451  df-kgen 17567
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