Theorem kgenidm 23462

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 23456	. . . 4 ⊢ 𝑘Gen:Top⟶Top
2		ffn 6651	. . . 4 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
3		fvelrnb 6882	. . . 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 22833	. . . . . . . . . . 11 ⊢ (𝑗 ∈ Top ↔ 𝑗 ∈ (TopOn‘∪ 𝑗))
6		kgentopon 23453	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (TopOn‘∪ 𝑗) → (𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗))
7	5, 6	sylbi 217	. . . . . . . . . 10 ⊢ (𝑗 ∈ Top → (𝑘Gen‘𝑗) ∈ (TopOn‘∪ 𝑗))
8		kgentopon 23453	. . . . . . . . . 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 22842	. . . . . . . . 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 23461	. . . . . . . . . . . . . 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 23452	. . . . . . . . . . . 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 23460	. . . . . . . . . . . 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 2834	. . . . . . . . . 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 3124	. . . . . . . 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 23451	. . . . . . . . 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 3935	. . . . 5 ⊢ (𝑗 ∈ Top → (𝑘Gen‘(𝑘Gen‘𝑗)) ⊆ (𝑘Gen‘𝑗))
30		fveq2 6822	. . . . . 6 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘(𝑘Gen‘𝑗)) = (𝑘Gen‘𝐽))
31		id 22	. . . . . 6 ⊢ ((𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝑗) = 𝐽)
32	30, 31	sseq12d 3963	. . . . 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 3126	. . 3 ⊢ (∃𝑗 ∈ Top (𝑘Gen‘𝑗) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
35	4, 34	sylbi 217	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
36		kgentop 23457	. . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
37		kgenss 23458	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
38	36, 37	syl 17	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ⊆ (𝑘Gen‘𝐽))
39	35, 38	eqssd 3947	1 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547  ∪ cuni 4856  ran crn 5615   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↾_t crest 17324  Topctop 22808  TopOnctopon 22825  Compccmp 23301  𝑘Genckgen 23448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-en 8870  df-fin 8873  df-fi 9295  df-rest 17326  df-topgen 17347  df-top 22809  df-topon 22826  df-bases 22861  df-cmp 23302  df-kgen 23449

This theorem is referenced by:  iskgen2  23463  kgencn3  23473  txkgen  23567