Theorem llyidm 21291

Description: Idempotence of the "locally" predicate, i.e. being "locally A " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
llyidm	|- Locally Locally A = Locally A

Proof of Theorem llyidm

Dummy variables j u v x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		llytop 21275	. . . 4 |- ( j e. Locally Locally A -> j e. Top )
2		llyi 21277	. . . . . . 7 |- ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) -> E. u e. j ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) )
3		simprr3 1111	. . . . . . . . 9 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> ( j ↾t u ) e. Locally A )
4		simprl 794	. . . . . . . . . 10 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> u e. j )
5		ssid 3624	. . . . . . . . . . 11 |- u C_ u
6	5	a1i 11	. . . . . . . . . 10 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> u C_ u )
7	1	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) -> j e. Top )
8	7	adantr 481	. . . . . . . . . . 11 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> j e. Top )
9		restopn2 20981	. . . . . . . . . . 11 |- ( ( j e. Top /\ u e. j ) -> ( u e. ( j ↾t u ) <-> ( u e. j /\ u C_ u ) ) )
10	8, 4, 9	syl2anc 693	. . . . . . . . . 10 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> ( u e. ( j ↾t u ) <-> ( u e. j /\ u C_ u ) ) )
11	4, 6, 10	mpbir2and 957	. . . . . . . . 9 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> u e. ( j ↾t u ) )
12		simprr2 1110	. . . . . . . . 9 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> y e. u )
13		llyi 21277	. . . . . . . . 9 |- ( ( ( j ↾t u ) e. Locally A /\ u e. ( j ↾t u ) /\ y e. u ) -> E. v e. ( j ↾t u ) ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) )
14	3, 11, 12, 13	syl3anc 1326	. . . . . . . 8 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> E. v e. ( j ↾t u ) ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) )
15		restopn2 20981	. . . . . . . . . . . 12 |- ( ( j e. Top /\ u e. j ) -> ( v e. ( j ↾t u ) <-> ( v e. j /\ v C_ u ) ) )
16	8, 4, 15	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> ( v e. ( j ↾t u ) <-> ( v e. j /\ v C_ u ) ) )
17		simpl 473	. . . . . . . . . . 11 |- ( ( v e. j /\ v C_ u ) -> v e. j )
18	16, 17	syl6bi 243	. . . . . . . . . 10 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> ( v e. ( j ↾t u ) -> v e. j ) )
19		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> v e. j )
20		simprr1 1109	. . . . . . . . . . . . . . 15 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> v C_ u )
21		simprr1 1109	. . . . . . . . . . . . . . . 16 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> u C_ x )
22	21	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> u C_ x )
23	20, 22	sstrd 3613	. . . . . . . . . . . . . 14 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> v C_ x )
24		selpw 4165	. . . . . . . . . . . . . 14 |- ( v e. ~P x <-> v C_ x )
25	23, 24	sylibr 224	. . . . . . . . . . . . 13 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> v e. ~P x )
26	19, 25	elind 3798	. . . . . . . . . . . 12 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> v e. ( j i^i ~P x ) )
27		simprr2 1110	. . . . . . . . . . . 12 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> y e. v )
28	8	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> j e. Top )
29		simplrl 800	. . . . . . . . . . . . . 14 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> u e. j )
30		restabs 20969	. . . . . . . . . . . . . 14 |- ( ( j e. Top /\ v C_ u /\ u e. j ) -> ( ( j ↾t u ) ↾t v ) = ( j ↾t v ) )
31	28, 20, 29, 30	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> ( ( j ↾t u ) ↾t v ) = ( j ↾t v ) )
32		simprr3 1111	. . . . . . . . . . . . 13 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> ( ( j ↾t u ) ↾t v ) e. A )
33	31, 32	eqeltrrd 2702	. . . . . . . . . . . 12 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> ( j ↾t v ) e. A )
34	26, 27, 33	jca32 558	. . . . . . . . . . 11 |- ( ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) /\ ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) ) -> ( v e. ( j i^i ~P x ) /\ ( y e. v /\ ( j ↾t v ) e. A ) ) )
35	34	ex 450	. . . . . . . . . 10 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> ( ( v e. j /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) -> ( v e. ( j i^i ~P x ) /\ ( y e. v /\ ( j ↾t v ) e. A ) ) ) )
36	18, 35	syland 498	. . . . . . . . 9 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> ( ( v e. ( j ↾t u ) /\ ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) ) -> ( v e. ( j i^i ~P x ) /\ ( y e. v /\ ( j ↾t v ) e. A ) ) ) )
37	36	reximdv2 3014	. . . . . . . 8 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> ( E. v e. ( j ↾t u ) ( v C_ u /\ y e. v /\ ( ( j ↾t u ) ↾t v ) e. A ) -> E. v e. ( j i^i ~P x ) ( y e. v /\ ( j ↾t v ) e. A ) ) )
38	14, 37	mpd 15	. . . . . . 7 |- ( ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) /\ ( u e. j /\ ( u C_ x /\ y e. u /\ ( j ↾t u ) e. Locally A ) ) ) -> E. v e. ( j i^i ~P x ) ( y e. v /\ ( j ↾t v ) e. A ) )
39	2, 38	rexlimddv 3035	. . . . . 6 |- ( ( j e. Locally Locally A /\ x e. j /\ y e. x ) -> E. v e. ( j i^i ~P x ) ( y e. v /\ ( j ↾t v ) e. A ) )
40	39	3expb 1266	. . . . 5 |- ( ( j e. Locally Locally A /\ ( x e. j /\ y e. x ) ) -> E. v e. ( j i^i ~P x ) ( y e. v /\ ( j ↾t v ) e. A ) )
41	40	ralrimivva 2971	. . . 4 |- ( j e. Locally Locally A -> A. x e. j A. y e. x E. v e. ( j i^i ~P x ) ( y e. v /\ ( j ↾t v ) e. A ) )
42		islly 21271	. . . 4 |- ( j e. Locally A <-> ( j e. Top /\ A. x e. j A. y e. x E. v e. ( j i^i ~P x ) ( y e. v /\ ( j ↾t v ) e. A ) ) )
43	1, 41, 42	sylanbrc 698	. . 3 |- ( j e. Locally Locally A -> j e. Locally A )
44	43	ssriv 3607	. 2 |- Locally Locally A C_ Locally A
45		llyrest 21288	. . . . 5 |- ( ( j e. Locally A /\ x e. j ) -> ( j ↾t x ) e. Locally A )
46	45	adantl 482	. . . 4 |- ( ( T. /\ ( j e. Locally A /\ x e. j ) ) -> ( j ↾t x ) e. Locally A )
47		llytop 21275	. . . . . 6 |- ( j e. Locally A -> j e. Top )
48	47	ssriv 3607	. . . . 5 |- Locally A C_ Top
49	48	a1i 11	. . . 4 |- ( T. -> Locally A C_ Top )
50	46, 49	restlly 21286	. . 3 |- ( T. -> Locally A C_ Locally Locally A )
51	50	trud 1493	. 2 |- Locally A C_ Locally Locally A
52	44, 51	eqssi 3619	1 |- Locally Locally A = Locally A

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  Locally clly 21267

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-lly 21269

This theorem is referenced by: (None)