Theorem kgenidm 23512

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 23506	. . . 4 ⊢ 𝑘Gen:Top⟶Top
2		ffn 6668	. . . 4 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
3		fvelrnb 6900	. . . 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 22883	. . . . . . . . . . 11 ⊢ (𝑗 ∈ Top ↔ 𝑗 ∈ (TopOn‘∪ 𝑗))
6		kgentopon 23503	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (TopOn‘∪ 𝑗) → (𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗))
7	5, 6	sylbi 217	. . . . . . . . . 10 ⊢ (𝑗 ∈ Top → (𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗))
8		kgentopon 23503	. . . . . . . . . 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 22892	. . . . . . . . 9 ⊢ (((𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘∪ 𝑗) ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ⊆ ∪ 𝑗)
11	9, 10	sylan 581	. . . . . . . 8 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ⊆ ∪ 𝑗)
12		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)))
13		kgencmp2 23511	. . . . . . . . . . . . . 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 750	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
16		kgeni 23502	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) ∧ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp) → (𝑥 ∩ 𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
17	12, 15, 16	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → (𝑥 ∩ 𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
18		kgencmp 23510	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ Top ∧ (𝑗 ↾t 𝑘) ∈ Comp) → (𝑗 ↾t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
19	18	ad2ant2rl 750	. . . . . . . . . . 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 2839	. . . . . . . . . 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 3129	. . . . . . . 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 23501	. . . . . . . . 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 714	. . . . . . 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 3927	. . . . 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 3955	. . . . 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 3131	. . 3 ⊢ (∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
35	4, 34	sylbi 217	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
36		kgentop 23507	. . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
37		kgenss 23508	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
38	36, 37	syl 17	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ⊆ (𝑘Gen‘𝐽))
39	35, 38	eqssd 3939	1 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850  ran crn 5632   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↾_t crest 17383  Topctop 22858  TopOnctopon 22875  Compccmp 23351  𝑘Genckgen 23498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-en 8894  df-fin 8897  df-fi 9324  df-rest 17385  df-topgen 17406  df-top 22859  df-topon 22876  df-bases 22911  df-cmp 23352  df-kgen 23499

This theorem is referenced by:  iskgen2  23513  kgencn3  23523  txkgen  23617