Theorem kgenidm 21350

Description: The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)

Assertion

Ref	Expression
kgenidm	|- ( J e. 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 21344	. . . 4 |- 𝑘Gen : Top --> Top
2		ffn 6045	. . . 4 |- (𝑘Gen : Top --> Top -> 𝑘Gen Fn Top )
3		fvelrnb 6243	. . . 4 |- (𝑘Gen Fn Top -> ( J e. ran 𝑘Gen <-> E. j e. Top (𝑘Gen ` j ) = J ) )
4	1, 2, 3	mp2b 10	. . 3 |- ( J e. ran 𝑘Gen <-> E. j e. Top (𝑘Gen ` j ) = J )
5		eqid 2622	. . . . . . . . . . . 12 |- U. j = U. j
6	5	toptopon 20722	. . . . . . . . . . 11 |- ( j e. Top <-> j e. (TopOn ` U. j ) )
7		kgentopon 21341	. . . . . . . . . . 11 |- ( j e. (TopOn ` U. j ) -> (𝑘Gen ` j ) e. (TopOn ` U. j ) )
8	6, 7	sylbi 207	. . . . . . . . . 10 |- ( j e. Top -> (𝑘Gen ` j ) e. (TopOn ` U. j ) )
9		kgentopon 21341	. . . . . . . . . 10 |- ( (𝑘Gen ` j ) e. (TopOn ` U. j ) -> (𝑘Gen ` (𝑘Gen ` j ) ) e. (TopOn ` U. j ) )
10	8, 9	syl 17	. . . . . . . . 9 |- ( j e. Top -> (𝑘Gen ` (𝑘Gen ` j ) ) e. (TopOn ` U. j ) )
11		toponss 20731	. . . . . . . . 9 |- ( ( (𝑘Gen ` (𝑘Gen ` j ) ) e. (TopOn ` U. j ) /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) -> x C_ U. j )
12	10, 11	sylan 488	. . . . . . . 8 |- ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) -> x C_ U. j )
13		simplr 792	. . . . . . . . . . . 12 |- ( ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) /\ ( k e. ~P U. j /\ ( j ↾t k ) e. Comp ) ) -> x e. (𝑘Gen ` (𝑘Gen ` j ) ) )
14		kgencmp2 21349	. . . . . . . . . . . . . 14 |- ( j e. Top -> ( ( j ↾t k ) e. Comp <-> ( (𝑘Gen ` j ) ↾t k ) e. Comp ) )
15	14	biimpa 501	. . . . . . . . . . . . 13 |- ( ( j e. Top /\ ( j ↾t k ) e. Comp ) -> ( (𝑘Gen ` j ) ↾t k ) e. Comp )
16	15	ad2ant2rl 785	. . . . . . . . . . . 12 |- ( ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) /\ ( k e. ~P U. j /\ ( j ↾t k ) e. Comp ) ) -> ( (𝑘Gen ` j ) ↾t k ) e. Comp )
17		kgeni 21340	. . . . . . . . . . . 12 |- ( ( x e. (𝑘Gen ` (𝑘Gen ` j ) ) /\ ( (𝑘Gen ` j ) ↾t k ) e. Comp ) -> ( x i^i k ) e. ( (𝑘Gen ` j ) ↾t k ) )
18	13, 16, 17	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) /\ ( k e. ~P U. j /\ ( j ↾t k ) e. Comp ) ) -> ( x i^i k ) e. ( (𝑘Gen ` j ) ↾t k ) )
19		kgencmp 21348	. . . . . . . . . . . 12 |- ( ( j e. Top /\ ( j ↾t k ) e. Comp ) -> ( j ↾t k ) = ( (𝑘Gen ` j ) ↾t k ) )
20	19	ad2ant2rl 785	. . . . . . . . . . 11 |- ( ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) /\ ( k e. ~P U. j /\ ( j ↾t k ) e. Comp ) ) -> ( j ↾t k ) = ( (𝑘Gen ` j ) ↾t k ) )
21	18, 20	eleqtrrd 2704	. . . . . . . . . 10 |- ( ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) /\ ( k e. ~P U. j /\ ( j ↾t k ) e. Comp ) ) -> ( x i^i k ) e. ( j ↾t k ) )
22	21	expr 643	. . . . . . . . 9 |- ( ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) /\ k e. ~P U. j ) -> ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) )
23	22	ralrimiva 2966	. . . . . . . 8 |- ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) -> A. k e. ~P U. j ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) )
24		simpl 473	. . . . . . . . . 10 |- ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) -> j e. Top )
25	24, 6	sylib 208	. . . . . . . . 9 |- ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) -> j e. (TopOn ` U. j ) )
26		elkgen 21339	. . . . . . . . 9 |- ( j e. (TopOn ` U. j ) -> ( x e. (𝑘Gen ` j ) <-> ( x C_ U. j /\ A. k e. ~P U. j ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) ) ) )
27	25, 26	syl 17	. . . . . . . 8 |- ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) -> ( x e. (𝑘Gen ` j ) <-> ( x C_ U. j /\ A. k e. ~P U. j ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) ) ) )
28	12, 23, 27	mpbir2and 957	. . . . . . 7 |- ( ( j e. Top /\ x e. (𝑘Gen ` (𝑘Gen ` j ) ) ) -> x e. (𝑘Gen ` j ) )
29	28	ex 450	. . . . . 6 |- ( j e. Top -> ( x e. (𝑘Gen ` (𝑘Gen ` j ) ) -> x e. (𝑘Gen ` j ) ) )
30	29	ssrdv 3609	. . . . 5 |- ( j e. Top -> (𝑘Gen ` (𝑘Gen ` j ) ) C_ (𝑘Gen ` j ) )
31		fveq2 6191	. . . . . 6 |- ( (𝑘Gen ` j ) = J -> (𝑘Gen ` (𝑘Gen ` j ) ) = (𝑘Gen ` J ) )
32		id 22	. . . . . 6 |- ( (𝑘Gen ` j ) = J -> (𝑘Gen ` j ) = J )
33	31, 32	sseq12d 3634	. . . . 5 |- ( (𝑘Gen ` j ) = J -> ( (𝑘Gen ` (𝑘Gen ` j ) ) C_ (𝑘Gen ` j ) <-> (𝑘Gen ` J ) C_ J ) )
34	30, 33	syl5ibcom 235	. . . 4 |- ( j e. Top -> ( (𝑘Gen ` j ) = J -> (𝑘Gen ` J ) C_ J ) )
35	34	rexlimiv 3027	. . 3 |- ( E. j e. Top (𝑘Gen ` j ) = J -> (𝑘Gen ` J ) C_ J )
36	4, 35	sylbi 207	. 2 |- ( J e. ran 𝑘Gen -> (𝑘Gen ` J ) C_ J )
37		kgentop 21345	. . 3 |- ( J e. ran 𝑘Gen -> J e. Top )
38		kgenss 21346	. . 3 |- ( J e. Top -> J C_ (𝑘Gen ` J ) )
39	37, 38	syl 17	. 2 |- ( J e. ran 𝑘Gen -> J C_ (𝑘Gen ` J ) )
40	36, 39	eqssd 3620	1 |- ( J e. ran 𝑘Gen -> (𝑘Gen ` J ) = J )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   Compccmp 21189  𝑘Genckgen 21336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-kgen 21337

This theorem is referenced by:  iskgen2  21351  kgencn3  21361  txkgen  21455