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Theorem kgenidm 23471
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 23465 . . . 4 𝑘Gen:Top⟶Top
2 ffn 6727 . . . 4 (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
3 fvelrnb 6964 . . . 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 22840 . . . . . . . . . . 11 (𝑗 ∈ Top ↔ 𝑗 ∈ (TopOn‘ 𝑗))
6 kgentopon 23462 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘ 𝑗) → (𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗))
75, 6sylbi 216 . . . . . . . . . 10 (𝑗 ∈ Top → (𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗))
8 kgentopon 23462 . . . . . . . . . 10 ((𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗) → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗))
97, 8syl 17 . . . . . . . . 9 (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗))
10 toponss 22849 . . . . . . . . 9 (((𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗) ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 𝑗)
119, 10sylan 578 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 𝑗)
12 simplr 767 . . . . . . . . . . . 12 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)))
13 kgencmp2 23470 . . . . . . . . . . . . . 14 (𝑗 ∈ Top → ((𝑗t 𝑘) ∈ Comp ↔ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp))
1413biimpa 475 . . . . . . . . . . . . 13 ((𝑗 ∈ Top ∧ (𝑗t 𝑘) ∈ Comp) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
1514ad2ant2rl 747 . . . . . . . . . . . 12 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
16 kgeni 23461 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) ∧ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp) → (𝑥𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
1712, 15, 16syl2anc 582 . . . . . . . . . . 11 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
18 kgencmp 23469 . . . . . . . . . . . 12 ((𝑗 ∈ Top ∧ (𝑗t 𝑘) ∈ Comp) → (𝑗t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
1918ad2ant2rl 747 . . . . . . . . . . 11 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑗t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
2017, 19eleqtrrd 2832 . . . . . . . . . 10 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ (𝑗t 𝑘))
2120expr 455 . . . . . . . . 9 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ 𝑘 ∈ 𝒫 𝑗) → ((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))
2221ralrimiva 3143 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))
23 simpl 481 . . . . . . . . . 10 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ Top)
2423, 5sylib 217 . . . . . . . . 9 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ (TopOn‘ 𝑗))
25 elkgen 23460 . . . . . . . . 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 711 . . . . . . 7 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ∈ (𝑘Gen‘𝑗))
2827ex 411 . . . . . 6 (𝑗 ∈ Top → (𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) → 𝑥 ∈ (𝑘Gen‘𝑗)))
2928ssrdv 3988 . . . . 5 (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗))
30 fveq2 6902 . . . . . 6 ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘(𝑘Gen‘𝑗)) = (𝑘Gen‘𝐽))
31 id 22 . . . . . 6 ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝑗) = 𝐽)
3230, 31sseq12d 4015 . . . . 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 3145 . . 3 (∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
354, 34sylbi 216 . 2 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
36 kgentop 23466 . . 3 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
37 kgenss 23467 . . 3 (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
3836, 37syl 17 . 2 (𝐽 ∈ ran 𝑘Gen → 𝐽 ⊆ (𝑘Gen‘𝐽))
3935, 38eqssd 3999 1 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  wrex 3067  cin 3948  wss 3949  𝒫 cpw 4606   cuni 4912  ran crn 5683   Fn wfn 6548  wf 6549  cfv 6553  (class class class)co 7426  t crest 17409  Topctop 22815  TopOnctopon 22832  Compccmp 23310  𝑘Genckgen 23457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-en 8971  df-fin 8974  df-fi 9442  df-rest 17411  df-topgen 17432  df-top 22816  df-topon 22833  df-bases 22869  df-cmp 23311  df-kgen 23458
This theorem is referenced by:  iskgen2  23472  kgencn3  23482  txkgen  23576
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