Theorem kgenidm 23673

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.

Step	Hyp	Ref	Expression
1		kgenf 23667	. . . 4 ⊢ 𝑘Gen:Top⟶Top
2		ffn 6706	. . . 4 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
3		fvelrnb 6942	. . . 4 ⊢ (𝑘Gen Fn Top → (𝐽 ∈ ran 𝑘Gen ↔ ∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽))
4	1, 2, 3	mp2b 10	. . 3 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ ∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽)
5		toptopon2 23044	. . . . . . . . . . 11 ⊢ (𝑗 ∈ Top ↔ 𝑗 ∈ (TopOn‘∪ 𝑗))
6		kgentopon 23664	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (TopOn‘∪ 𝑗) → (𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗))
7	5, 6	sylbi 220	. . . . . . . . . 10 ⊢ (𝑗 ∈ Top → (𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗))
8		kgentopon 23664	. . . . . . . . . 10 ⊢ ((𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗) → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘∪ 𝑗))
9	7, 8	syl 18	. . . . . . . . 9 ⊢ (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘∪ 𝑗))
10		toponss 23053	. . . . . . . . 9 ⊢ (((𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘∪ 𝑗) ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ⊆ ∪ 𝑗)
11	9, 10	sylan 591	. . . . . . . 8 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ⊆ ∪ 𝑗)
12		simplr 780	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)))
13		kgencmp2 23672	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ Top → ((𝑗 ↾t 𝑘) ∈ Comp ↔ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp))
14	13	biimpa 481	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ Top ∧ (𝑗 ↾t 𝑘) ∈ Comp) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
15	14	ad2ant2rl 761	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
16		kgeni 23663	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) ∧ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp) → (𝑥 ∩ 𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
17	12, 15, 16	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → (𝑥 ∩ 𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
18		kgencmp 23671	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ Top ∧ (𝑗 ↾t 𝑘) ∈ Comp) → (𝑗 ↾t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
19	18	ad2ant2rl 761	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → (𝑗 ↾t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
20	17, 19	eleqtrrd 2872	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))
21	20	expr 461	. . . . . . . . 9 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ 𝑘 ∈ 𝒫 ∪ 𝑗) → ((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)))
22	21	ralrimiva 3163	. . . . . . . 8 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)))
23	5	birani 508	. . . . . . . . 9 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ (TopOn‘∪ 𝑗))
24		elkgen 23662	. . . . . . . . 9 ⊢ (𝑗 ∈ (TopOn‘∪ 𝑗) → (𝑥 ∈ (𝑘Gen‘𝑗) ↔ (𝑥 ⊆ ∪ 𝑗 ∧ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)))))
25	23, 24	syl 18	. . . . . . . 8 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → (𝑥 ∈ (𝑘Gen‘𝑗) ↔ (𝑥 ⊆ ∪ 𝑗 ∧ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)))))
26	11, 22, 25	mpbir2and 725	. . . . . . 7 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ∈ (𝑘Gen‘𝑗))
27	26	ex 417	. . . . . 6 ⊢ (𝑗 ∈ Top → (𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) → 𝑥 ∈ (𝑘Gen‘𝑗)))
28	27	ssrdv 3951	. . . . 5 ⊢ (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗))
29		fveq2 6882	. . . . . 6 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘(𝑘Gen‘𝑗)) = (𝑘Gen‘𝐽))
30		id 23	. . . . . 6 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝑗) = 𝐽)
31	29, 30	sseq12d 3978	. . . . 5 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → ((𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗) ↔ (𝑘Gen‘𝐽) ⊆ 𝐽))
32	28, 31	syl5ibcom 248	. . . 4 ⊢ (𝑗 ∈ Top → ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽))
33	32	rexlimiv 3165	. . 3 ⊢ (∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
34	4, 33	sylbi 220	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
35		kgentop 23668	. . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
36		kgenss 23669	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
37	35, 36	syl 18	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ⊆ (𝑘Gen‘𝐽))
38	34, 37	eqssd 3962	1 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4876  ran crn 5663   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ↾_t crest 17473  Topctop 23019  TopOnctopon 23036  Compccmp 23512  𝑘Genckgen 23659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-en 8944  df-fin 8947  df-fi 9371  df-rest 17475  df-topgen 17496  df-top 23020  df-topon 23037  df-bases 23072  df-cmp 23513  df-kgen 23660

This theorem is referenced by:  iskgen2  23674  kgencn3  23684  txkgen  23778