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Theorem kgenidm 21398
Description: The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgenidm (𝐽 ∈ 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 21392 . . . 4 𝑘Gen:Top⟶Top
2 ffn 6083 . . . 4 (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
3 fvelrnb 6282 . . . 4 (𝑘Gen Fn Top → (𝐽 ∈ ran 𝑘Gen ↔ ∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽))
41, 2, 3mp2b 10 . . 3 (𝐽 ∈ ran 𝑘Gen ↔ ∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽)
5 eqid 2651 . . . . . . . . . . . 12 𝑗 = 𝑗
65toptopon 20770 . . . . . . . . . . 11 (𝑗 ∈ Top ↔ 𝑗 ∈ (TopOn‘ 𝑗))
7 kgentopon 21389 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘ 𝑗) → (𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗))
86, 7sylbi 207 . . . . . . . . . 10 (𝑗 ∈ Top → (𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗))
9 kgentopon 21389 . . . . . . . . . 10 ((𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗) → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗))
108, 9syl 17 . . . . . . . . 9 (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗))
11 toponss 20779 . . . . . . . . 9 (((𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗) ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 𝑗)
1210, 11sylan 487 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 𝑗)
13 simplr 807 . . . . . . . . . . . 12 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)))
14 kgencmp2 21397 . . . . . . . . . . . . . 14 (𝑗 ∈ Top → ((𝑗t 𝑘) ∈ Comp ↔ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp))
1514biimpa 500 . . . . . . . . . . . . 13 ((𝑗 ∈ Top ∧ (𝑗t 𝑘) ∈ Comp) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
1615ad2ant2rl 800 . . . . . . . . . . . 12 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
17 kgeni 21388 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) ∧ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp) → (𝑥𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
1813, 16, 17syl2anc 694 . . . . . . . . . . 11 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
19 kgencmp 21396 . . . . . . . . . . . 12 ((𝑗 ∈ Top ∧ (𝑗t 𝑘) ∈ Comp) → (𝑗t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
2019ad2ant2rl 800 . . . . . . . . . . 11 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑗t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
2118, 20eleqtrrd 2733 . . . . . . . . . 10 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ (𝑗t 𝑘))
2221expr 642 . . . . . . . . 9 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ 𝑘 ∈ 𝒫 𝑗) → ((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))
2322ralrimiva 2995 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))
24 simpl 472 . . . . . . . . . 10 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ Top)
2524, 6sylib 208 . . . . . . . . 9 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ (TopOn‘ 𝑗))
26 elkgen 21387 . . . . . . . . 9 (𝑗 ∈ (TopOn‘ 𝑗) → (𝑥 ∈ (𝑘Gen‘𝑗) ↔ (𝑥 𝑗 ∧ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))))
2725, 26syl 17 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → (𝑥 ∈ (𝑘Gen‘𝑗) ↔ (𝑥 𝑗 ∧ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))))
2812, 23, 27mpbir2and 977 . . . . . . 7 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ∈ (𝑘Gen‘𝑗))
2928ex 449 . . . . . 6 (𝑗 ∈ Top → (𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) → 𝑥 ∈ (𝑘Gen‘𝑗)))
3029ssrdv 3642 . . . . 5 (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗))
31 fveq2 6229 . . . . . 6 ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘(𝑘Gen‘𝑗)) = (𝑘Gen‘𝐽))
32 id 22 . . . . . 6 ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝑗) = 𝐽)
3331, 32sseq12d 3667 . . . . 5 ((𝑘Gen‘𝑗) = 𝐽 → ((𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗) ↔ (𝑘Gen‘𝐽) ⊆ 𝐽))
3430, 33syl5ibcom 235 . . . 4 (𝑗 ∈ Top → ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽))
3534rexlimiv 3056 . . 3 (∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
364, 35sylbi 207 . 2 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
37 kgentop 21393 . . 3 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
38 kgenss 21394 . . 3 (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
3937, 38syl 17 . 2 (𝐽 ∈ ran 𝑘Gen → 𝐽 ⊆ (𝑘Gen‘𝐽))
4036, 39eqssd 3653 1 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  t crest 16128  Topctop 20746  TopOnctopon 20763  Compccmp 21237  𝑘Genckgen 21384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-kgen 21385
This theorem is referenced by:  iskgen2  21399  kgencn3  21409  txkgen  21503
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