Theorem llynlly 21280

Description: A locally A space is n-locally A: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
llynlly	|- ( J e. Locally A -> J e. 𝑛Locally A )

Proof of Theorem llynlly

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

Step	Hyp	Ref	Expression
1		llytop 21275	. 2 |- ( J e. Locally A -> J e. Top )
2		llyi 21277	. . . . 5 |- ( ( J e. Locally A /\ x e. J /\ y e. x ) -> E. u e. J ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) )
3		simpl1 1064	. . . . . . . . . . 11 |- ( ( ( J e. Locally A /\ x e. J /\ y e. x ) /\ ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) ) -> J e. Locally A )
4	3, 1	syl 17	. . . . . . . . . 10 |- ( ( ( J e. Locally A /\ x e. J /\ y e. x ) /\ ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) ) -> J e. Top )
5		simprl 794	. . . . . . . . . 10 |- ( ( ( J e. Locally A /\ x e. J /\ y e. x ) /\ ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) ) -> u e. J )
6		simprr2 1110	. . . . . . . . . 10 |- ( ( ( J e. Locally A /\ x e. J /\ y e. x ) /\ ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) ) -> y e. u )
7		opnneip 20923	. . . . . . . . . 10 |- ( ( J e. Top /\ u e. J /\ y e. u ) -> u e. ( ( nei ` J ) ` { y } ) )
8	4, 5, 6, 7	syl3anc 1326	. . . . . . . . 9 |- ( ( ( J e. Locally A /\ x e. J /\ y e. x ) /\ ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) ) -> u e. ( ( nei ` J ) ` { y } ) )
9		simprr1 1109	. . . . . . . . . 10 |- ( ( ( J e. Locally A /\ x e. J /\ y e. x ) /\ ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) ) -> u C_ x )
10		selpw 4165	. . . . . . . . . 10 |- ( u e. ~P x <-> u C_ x )
11	9, 10	sylibr 224	. . . . . . . . 9 |- ( ( ( J e. Locally A /\ x e. J /\ y e. x ) /\ ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) ) -> u e. ~P x )
12	8, 11	elind 3798	. . . . . . . 8 |- ( ( ( J e. Locally A /\ x e. J /\ y e. x ) /\ ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) ) -> u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) )
13		simprr3 1111	. . . . . . . 8 |- ( ( ( J e. Locally A /\ x e. J /\ y e. x ) /\ ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) ) -> ( J ↾t u ) e. A )
14	12, 13	jca 554	. . . . . . 7 |- ( ( ( J e. Locally A /\ x e. J /\ y e. x ) /\ ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) ) -> ( u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) /\ ( J ↾t u ) e. A ) )
15	14	ex 450	. . . . . 6 |- ( ( J e. Locally A /\ x e. J /\ y e. x ) -> ( ( u e. J /\ ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) ) -> ( u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) /\ ( J ↾t u ) e. A ) ) )
16	15	reximdv2 3014	. . . . 5 |- ( ( J e. Locally A /\ x e. J /\ y e. x ) -> ( E. u e. J ( u C_ x /\ y e. u /\ ( J ↾t u ) e. A ) -> E. u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) ( J ↾t u ) e. A ) )
17	2, 16	mpd 15	. . . 4 |- ( ( J e. Locally A /\ x e. J /\ y e. x ) -> E. u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) ( J ↾t u ) e. A )
18	17	3expb 1266	. . 3 |- ( ( J e. Locally A /\ ( x e. J /\ y e. x ) ) -> E. u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) ( J ↾t u ) e. A )
19	18	ralrimivva 2971	. 2 |- ( J e. Locally A -> A. x e. J A. y e. x E. u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) ( J ↾t u ) e. A )
20		isnlly 21272	. 2 |- ( J e. 𝑛Locally A <-> ( J e. Top /\ A. x e. J A. y e. x E. u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) ( J ↾t u ) e. A ) )
21	1, 19, 20	sylanbrc 698	1 |- ( J e. Locally A -> J e. 𝑛Locally A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698   neicnei 20901  Locally clly 21267  𝑛Locally cnlly 21268

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-top 20699  df-nei 20902  df-lly 21269  df-nlly 21270

This theorem is referenced by:  llyssnlly  21281  symgtgp  21905