Theorem kgenidm 23467

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 23461	. . . 4 ⊢ 𝑘Gen:Top⟶Top
2		ffn 6670	. . . 4 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
3		fvelrnb 6903	. . . 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 22838	. . . . . . . . . . 11 ⊢ (𝑗 ∈ Top ↔ 𝑗 ∈ (TopOn‘∪ 𝑗))
6		kgentopon 23458	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (TopOn‘∪ 𝑗) → (𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗))
7	5, 6	sylbi 217	. . . . . . . . . 10 ⊢ (𝑗 ∈ Top → (𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗))
8		kgentopon 23458	. . . . . . . . . 10 ⊢ ((𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗) → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘∪ 𝑗))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘∪ 𝑗))
10		toponss 22847	. . . . . . . . 9 ⊢ (((𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘∪ 𝑗) ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ⊆ ∪ 𝑗)
11	9, 10	sylan 580	. . . . . . . 8 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ⊆ ∪ 𝑗)
12		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)))
13		kgencmp2 23466	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ Top → ((𝑗 ↾t 𝑘) ∈ Comp ↔ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp))
14	13	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ Top ∧ (𝑗 ↾t 𝑘) ∈ Comp) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
15	14	ad2ant2rl 749	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
16		kgeni 23457	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) ∧ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp) → (𝑥 ∩ 𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
17	12, 15, 16	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → (𝑥 ∩ 𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
18		kgencmp 23465	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ Top ∧ (𝑗 ↾t 𝑘) ∈ Comp) → (𝑗 ↾t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
19	18	ad2ant2rl 749	. . . . . . . . . . 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 2831	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))
21	20	expr 456	. . . . . . . . 9 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ 𝑘 ∈ 𝒫 ∪ 𝑗) → ((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)))
22	21	ralrimiva 3125	. . . . . . . 8 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)))
23		simpl 482	. . . . . . . . . 10 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ Top)
24	23, 5	sylib 218	. . . . . . . . 9 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ (TopOn‘∪ 𝑗))
25		elkgen 23456	. . . . . . . . 9 ⊢ (𝑗 ∈ (TopOn‘∪ 𝑗) → (𝑥 ∈ (𝑘Gen‘𝑗) ↔ (𝑥 ⊆ ∪ 𝑗 ∧ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)))))
26	24, 25	syl 17	. . . . . . . 8 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → (𝑥 ∈ (𝑘Gen‘𝑗) ↔ (𝑥 ⊆ ∪ 𝑗 ∧ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)))))
27	11, 22, 26	mpbir2and 713	. . . . . . 7 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ∈ (𝑘Gen‘𝑗))
28	27	ex 412	. . . . . 6 ⊢ (𝑗 ∈ Top → (𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) → 𝑥 ∈ (𝑘Gen‘𝑗)))
29	28	ssrdv 3949	. . . . 5 ⊢ (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗))
30		fveq2 6840	. . . . . 6 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘(𝑘Gen‘𝑗)) = (𝑘Gen‘𝐽))
31		id 22	. . . . . 6 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝑗) = 𝐽)
32	30, 31	sseq12d 3977	. . . . 5 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → ((𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗) ↔ (𝑘Gen‘𝐽) ⊆ 𝐽))
33	29, 32	syl5ibcom 245	. . . 4 ⊢ (𝑗 ∈ Top → ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽))
34	33	rexlimiv 3127	. . 3 ⊢ (∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
35	4, 34	sylbi 217	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
36		kgentop 23462	. . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
37		kgenss 23463	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
38	36, 37	syl 17	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ⊆ (𝑘Gen‘𝐽))
39	35, 38	eqssd 3961	1 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867  ran crn 5632   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ↾_t crest 17359  Topctop 22813  TopOnctopon 22830  Compccmp 23306  𝑘Genckgen 23453

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-en 8896  df-fin 8899  df-fi 9338  df-rest 17361  df-topgen 17382  df-top 22814  df-topon 22831  df-bases 22866  df-cmp 23307  df-kgen 23454

This theorem is referenced by:  iskgen2  23468  kgencn3  23478  txkgen  23572