Theorem kgenidm 22154

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 22148	. . . 4 ⊢ 𝑘Gen:Top⟶Top
2		ffn 6513	. . . 4 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
3		fvelrnb 6725	. . . 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 21525	. . . . . . . . . . 11 ⊢ (𝑗 ∈ Top ↔ 𝑗 ∈ (TopOn‘∪ 𝑗))
6		kgentopon 22145	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (TopOn‘∪ 𝑗) → (𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗))
7	5, 6	sylbi 219	. . . . . . . . . 10 ⊢ (𝑗 ∈ Top → (𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗))
8		kgentopon 22145	. . . . . . . . . 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 21534	. . . . . . . . 9 ⊢ (((𝑘Gen‘(𝑘Gen‘𝑗)) ∈ (TopOn‘∪ 𝑗) ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ⊆ ∪ 𝑗)
11	9, 10	sylan 582	. . . . . . . 8 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ⊆ ∪ 𝑗)
12		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)))
13		kgencmp2 22153	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ Top → ((𝑗 ↾t 𝑘) ∈ Comp ↔ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp))
14	13	biimpa 479	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ Top ∧ (𝑗 ↾t 𝑘) ∈ Comp) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
15	14	ad2ant2rl 747	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp)
16		kgeni 22144	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) ∧ ((𝑘Gen‘𝑗) ↾t 𝑘) ∈ Comp) → (𝑥 ∩ 𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
17	12, 15, 16	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → (𝑥 ∩ 𝑘) ∈ ((𝑘Gen‘𝑗) ↾t 𝑘))
18		kgencmp 22152	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ Top ∧ (𝑗 ↾t 𝑘) ∈ Comp) → (𝑗 ↾t 𝑘) = ((𝑘Gen‘𝑗) ↾t 𝑘))
19	18	ad2ant2rl 747	. . . . . . . . . . 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 2916	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑗 ∧ (𝑗 ↾t 𝑘) ∈ Comp)) → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))
21	20	expr 459	. . . . . . . . 9 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) ∧ 𝑘 ∈ 𝒫 ∪ 𝑗) → ((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)))
22	21	ralrimiva 3182	. . . . . . . 8 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)))
23		simpl 485	. . . . . . . . . 10 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ Top)
24	23, 5	sylib 220	. . . . . . . . 9 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑗 ∈ (TopOn‘∪ 𝑗))
25		elkgen 22143	. . . . . . . . 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 711	. . . . . . 7 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗))) → 𝑥 ∈ (𝑘Gen‘𝑗))
28	27	ex 415	. . . . . 6 ⊢ (𝑗 ∈ Top → (𝑥 ∈ (𝑘Gen‘(𝑘Gen‘𝑗)) → 𝑥 ∈ (𝑘Gen‘𝑗)))
29	28	ssrdv 3972	. . . . 5 ⊢ (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗))
30		fveq2 6669	. . . . . 6 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘(𝑘Gen‘𝑗)) = (𝑘Gen‘𝐽))
31		id 22	. . . . . 6 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝑗) = 𝐽)
32	30, 31	sseq12d 3999	. . . . 5 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → ((𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗) ↔ (𝑘Gen‘𝐽) ⊆ 𝐽))
33	29, 32	syl5ibcom 247	. . . 4 ⊢ (𝑗 ∈ Top → ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽))
34	33	rexlimiv 3280	. . 3 ⊢ (∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
35	4, 34	sylbi 219	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
36		kgentop 22149	. . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
37		kgenss 22150	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
38	36, 37	syl 17	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ⊆ (𝑘Gen‘𝐽))
39	35, 38	eqssd 3983	1 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ∩ cin 3934   ⊆ wss 3935  𝒫 cpw 4538  ∪ cuni 4837  ran crn 5555   Fn wfn 6349  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155   ↾_t crest 16693  Topctop 21500  TopOnctopon 21517  Compccmp 21993  𝑘Genckgen 22140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-oadd 8105  df-er 8288  df-en 8509  df-fin 8512  df-fi 8874  df-rest 16695  df-topgen 16716  df-top 21501  df-topon 21518  df-bases 21553  df-cmp 21994  df-kgen 22141

This theorem is referenced by:  iskgen2  22155  kgencn3  22165  txkgen  22259