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Theorem kgenidm 21722
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 21716 . . . 4 𝑘Gen:Top⟶Top
2 ffn 6279 . . . 4 (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
3 fvelrnb 6491 . . . 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 2826 . . . . . . . . . . . 12 𝑗 = 𝑗
65toptopon 21093 . . . . . . . . . . 11 (𝑗 ∈ Top ↔ 𝑗 ∈ (TopOn‘ 𝑗))
7 kgentopon 21713 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘ 𝑗) → (𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗))
86, 7sylbi 209 . . . . . . . . . 10 (𝑗 ∈ Top → (𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗))
9 kgentopon 21713 . . . . . . . . . 10 ((𝑘Gen‘𝑗) ∈ (TopOn‘ 𝑗) → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗))
108, 9syl 17 . . . . . . . . 9 (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗))
11 toponss 21103 . . . . . . . . 9 (((𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘ 𝑗) ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 𝑗)
1210, 11sylan 577 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 𝑗)
13 simplr 787 . . . . . . . . . . . 12 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)))
14 kgencmp2 21721 . . . . . . . . . . . . . 14 (𝑗 ∈ Top → ((𝑗t 𝑘) ∈ Comp ↔ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp))
1514biimpa 470 . . . . . . . . . . . . 13 ((𝑗 ∈ Top ∧ (𝑗t 𝑘) ∈ Comp) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
1615ad2ant2rl 757 . . . . . . . . . . . 12 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
17 kgeni 21712 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) ∧ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp) → (𝑥𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
1813, 16, 17syl2anc 581 . . . . . . . . . . 11 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
19 kgencmp 21720 . . . . . . . . . . . 12 ((𝑗 ∈ Top ∧ (𝑗t 𝑘) ∈ Comp) → (𝑗t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
2019ad2ant2rl 757 . . . . . . . . . . 11 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑗t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
2118, 20eleqtrrd 2910 . . . . . . . . . 10 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 𝑗 ∧ (𝑗t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ (𝑗t 𝑘))
2221expr 450 . . . . . . . . 9 (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ 𝑘 ∈ 𝒫 𝑗) → ((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))
2322ralrimiva 3176 . . . . . . . 8 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)))
24 simpl 476 . . . . . . . . . 10 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ Top)
2524, 6sylib 210 . . . . . . . . 9 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ (TopOn‘ 𝑗))
26 elkgen 21711 . . . . . . . . 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 706 . . . . . . 7 ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ∈ (𝑘Gen‘𝑗))
2928ex 403 . . . . . 6 (𝑗 ∈ Top → (𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) → 𝑥 ∈ (𝑘Gen‘𝑗)))
3029ssrdv 3834 . . . . 5 (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗))
31 fveq2 6434 . . . . . 6 ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘(𝑘Gen‘𝑗)) = (𝑘Gen‘𝐽))
32 id 22 . . . . . 6 ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝑗) = 𝐽)
3331, 32sseq12d 3860 . . . . 5 ((𝑘Gen‘𝑗) = 𝐽 → ((𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗) ↔ (𝑘Gen‘𝐽) ⊆ 𝐽))
3430, 33syl5ibcom 237 . . . 4 (𝑗 ∈ Top → ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽))
3534rexlimiv 3237 . . 3 (∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
364, 35sylbi 209 . 2 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
37 kgentop 21717 . . 3 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
38 kgenss 21718 . . 3 (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
3937, 38syl 17 . 2 (𝐽 ∈ ran 𝑘Gen → 𝐽 ⊆ (𝑘Gen‘𝐽))
4036, 39eqssd 3845 1 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wral 3118  wrex 3119  cin 3798  wss 3799  𝒫 cpw 4379   cuni 4659  ran crn 5344   Fn wfn 6119  wf 6120  cfv 6124  (class class class)co 6906  t crest 16435  Topctop 21069  TopOnctopon 21086  Compccmp 21561  𝑘Genckgen 21708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-oadd 7831  df-er 8010  df-en 8224  df-fin 8227  df-fi 8587  df-rest 16437  df-topgen 16458  df-top 21070  df-topon 21087  df-bases 21122  df-cmp 21562  df-kgen 21709
This theorem is referenced by:  iskgen2  21723  kgencn3  21733  txkgen  21827
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