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Theorem kgenidm 22687
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 22681 . . . 4 𝑘Gen:Top⟶Top
2 ffn 6594 . . . 4 (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
3 fvelrnb 6824 . . . 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 toptopon2 22056 . . . . . . . . . . 11 (𝑗 ∈ Top ↔ 𝑗 ∈ (TopOn‘ 𝑗))
6 kgentopon 22678 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘ 𝑗) → (𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗))
75, 6sylbi 216 . . . . . . . . . 10 (𝑗 ∈ Top → (𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗))
8 kgentopon 22678 . . . . . . . . . 10 ((𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗) → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗))
97, 8syl 17 . . . . . . . . 9 (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗))
10 toponss 22065 . . . . . . . . 9 (((𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗) ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 𝑗)
119, 10sylan 580 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 𝑗)
12 simplr 766 . . . . . . . . . . . 12 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)))
13 kgencmp2 22686 . . . . . . . . . . . . . 14 (𝑗 ∈ Top → ((𝑗t 𝑘) ∈ Comp ↔ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp))
1413biimpa 477 . . . . . . . . . . . . 13 ((𝑗 ∈ Top ∧ (𝑗t 𝑘) ∈ Comp) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
1514ad2ant2rl 746 . . . . . . . . . . . 12 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
16 kgeni 22677 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) ∧ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp) → (𝑥𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
1712, 15, 16syl2anc 584 . . . . . . . . . . 11 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
18 kgencmp 22685 . . . . . . . . . . . 12 ((𝑗 ∈ Top ∧ (𝑗t 𝑘) ∈ Comp) → (𝑗t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
1918ad2ant2rl 746 . . . . . . . . . . 11 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑗t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
2017, 19eleqtrrd 2842 . . . . . . . . . 10 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ (𝑗t 𝑘))
2120expr 457 . . . . . . . . 9 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ 𝑘 ∈ 𝒫 𝑗) → ((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))
2221ralrimiva 3103 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))
23 simpl 483 . . . . . . . . . 10 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ Top)
2423, 5sylib 217 . . . . . . . . 9 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ (TopOn‘ 𝑗))
25 elkgen 22676 . . . . . . . . 9 (𝑗 ∈ (TopOn‘ 𝑗) → (𝑥 ∈ (𝑘Gen‘𝑗) ↔ (𝑥 𝑗 ∧ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))))
2624, 25syl 17 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → (𝑥 ∈ (𝑘Gen‘𝑗) ↔ (𝑥 𝑗 ∧ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))))
2711, 22, 26mpbir2and 710 . . . . . . 7 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ∈ (𝑘Gen‘𝑗))
2827ex 413 . . . . . 6 (𝑗 ∈ Top → (𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) → 𝑥 ∈ (𝑘Gen‘𝑗)))
2928ssrdv 3928 . . . . 5 (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗))
30 fveq2 6768 . . . . . 6 ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘(𝑘Gen‘𝑗)) = (𝑘Gen‘𝐽))
31 id 22 . . . . . 6 ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝑗) = 𝐽)
3230, 31sseq12d 3955 . . . . 5 ((𝑘Gen‘𝑗) = 𝐽 → ((𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗) ↔ (𝑘Gen‘𝐽) ⊆ 𝐽))
3329, 32syl5ibcom 244 . . . 4 (𝑗 ∈ Top → ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽))
3433rexlimiv 3208 . . 3 (∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
354, 34sylbi 216 . 2 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
36 kgentop 22682 . . 3 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
37 kgenss 22683 . . 3 (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
3836, 37syl 17 . 2 (𝐽 ∈ ran 𝑘Gen → 𝐽 ⊆ (𝑘Gen‘𝐽))
3935, 38eqssd 3939 1 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  wrex 3065  cin 3887  wss 3888  𝒫 cpw 4535   cuni 4841  ran crn 5587   Fn wfn 6423  wf 6424  cfv 6428  (class class class)co 7269  t crest 17120  Topctop 22031  TopOnctopon 22048  Compccmp 22526  𝑘Genckgen 22673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5210  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-int 4882  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5486  df-eprel 5492  df-po 5500  df-so 5501  df-fr 5541  df-we 5543  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-ord 6264  df-on 6265  df-lim 6266  df-suc 6267  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-1st 7822  df-2nd 7823  df-en 8723  df-fin 8726  df-fi 9159  df-rest 17122  df-topgen 17143  df-top 22032  df-topon 22049  df-bases 22085  df-cmp 22527  df-kgen 22674
This theorem is referenced by:  iskgen2  22688  kgencn3  22698  txkgen  22792
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