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Theorem kgenidm 23555
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 23549 . . . 4 𝑘Gen:Top⟶Top
2 ffn 6736 . . . 4 (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
3 fvelrnb 6969 . . . 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 22924 . . . . . . . . . . 11 (𝑗 ∈ Top ↔ 𝑗 ∈ (TopOn‘ 𝑗))
6 kgentopon 23546 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘ 𝑗) → (𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗))
75, 6sylbi 217 . . . . . . . . . 10 (𝑗 ∈ Top → (𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗))
8 kgentopon 23546 . . . . . . . . . 10 ((𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗) → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗))
97, 8syl 17 . . . . . . . . 9 (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗))
10 toponss 22933 . . . . . . . . 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 769 . . . . . . . . . . . 12 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)))
13 kgencmp2 23554 . . . . . . . . . . . . . 14 (𝑗 ∈ Top → ((𝑗t 𝑘) ∈ Comp ↔ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp))
1413biimpa 476 . . . . . . . . . . . . 13 ((𝑗 ∈ Top ∧ (𝑗t 𝑘) ∈ Comp) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
1514ad2ant2rl 749 . . . . . . . . . . . 12 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
16 kgeni 23545 . . . . . . . . . . . 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 23553 . . . . . . . . . . . 12 ((𝑗 ∈ Top ∧ (𝑗t 𝑘) ∈ Comp) → (𝑗t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
1918ad2ant2rl 749 . . . . . . . . . . 11 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑗t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
2017, 19eleqtrrd 2844 . . . . . . . . . 10 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ (𝑗t 𝑘))
2120expr 456 . . . . . . . . 9 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ 𝑘 ∈ 𝒫 𝑗) → ((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))
2221ralrimiva 3146 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))
23 simpl 482 . . . . . . . . . 10 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ Top)
2423, 5sylib 218 . . . . . . . . 9 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ (TopOn‘ 𝑗))
25 elkgen 23544 . . . . . . . . 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 713 . . . . . . 7 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ∈ (𝑘Gen‘𝑗))
2827ex 412 . . . . . 6 (𝑗 ∈ Top → (𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) → 𝑥 ∈ (𝑘Gen‘𝑗)))
2928ssrdv 3989 . . . . 5 (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗))
30 fveq2 6906 . . . . . 6 ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘(𝑘Gen‘𝑗)) = (𝑘Gen‘𝐽))
31 id 22 . . . . . 6 ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝑗) = 𝐽)
3230, 31sseq12d 4017 . . . . 5 ((𝑘Gen‘𝑗) = 𝐽 → ((𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗) ↔ (𝑘Gen‘𝐽) ⊆ 𝐽))
3329, 32syl5ibcom 245 . . . 4 (𝑗 ∈ Top → ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽))
3433rexlimiv 3148 . . 3 (∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
354, 34sylbi 217 . 2 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
36 kgentop 23550 . . 3 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
37 kgenss 23551 . . 3 (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
3836, 37syl 17 . 2 (𝐽 ∈ ran 𝑘Gen → 𝐽 ⊆ (𝑘Gen‘𝐽))
3935, 38eqssd 4001 1 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  TopOnctopon 22916  Compccmp 23394  𝑘Genckgen 23541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-kgen 23542
This theorem is referenced by:  iskgen2  23556  kgencn3  23566  txkgen  23660
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