Theorem restlly 21286

Description: If the property A passes to open subspaces, then a space which is A is also locally A. (Contributed by Mario Carneiro, 2-Mar-2015.)

Hypotheses

Ref	Expression
restlly.1	|- ( ( ph /\ ( j e. A /\ x e. j ) ) -> ( j ↾t x ) e. A )
restlly.2	|- ( ph -> A C_ Top )

Assertion

Ref	Expression
restlly	|- ( ph -> A C_ Locally A )

Distinct variable groups:   x, j, A   ph, j, x

Proof of Theorem restlly

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

Step	Hyp	Ref	Expression
1		restlly.2	. . . . 5 |- ( ph -> A C_ Top )
2	1	sselda 3603	. . . 4 |- ( ( ph /\ j e. A ) -> j e. Top )
3		simprl 794	. . . . . . 7 |- ( ( ( ph /\ j e. A ) /\ ( x e. j /\ y e. x ) ) -> x e. j )
4		vex 3203	. . . . . . . . 9 |- x e. _V
5	4	pwid 4174	. . . . . . . 8 |- x e. ~P x
6	5	a1i 11	. . . . . . 7 |- ( ( ( ph /\ j e. A ) /\ ( x e. j /\ y e. x ) ) -> x e. ~P x )
7	3, 6	elind 3798	. . . . . 6 |- ( ( ( ph /\ j e. A ) /\ ( x e. j /\ y e. x ) ) -> x e. ( j i^i ~P x ) )
8		simprr 796	. . . . . 6 |- ( ( ( ph /\ j e. A ) /\ ( x e. j /\ y e. x ) ) -> y e. x )
9		restlly.1	. . . . . . . 8 |- ( ( ph /\ ( j e. A /\ x e. j ) ) -> ( j ↾t x ) e. A )
10	9	anassrs 680	. . . . . . 7 |- ( ( ( ph /\ j e. A ) /\ x e. j ) -> ( j ↾t x ) e. A )
11	10	adantrr 753	. . . . . 6 |- ( ( ( ph /\ j e. A ) /\ ( x e. j /\ y e. x ) ) -> ( j ↾t x ) e. A )
12		elequ2 2004	. . . . . . . 8 |- ( u = x -> ( y e. u <-> y e. x ) )
13		oveq2 6658	. . . . . . . . 9 |- ( u = x -> ( j ↾t u ) = ( j ↾t x ) )
14	13	eleq1d 2686	. . . . . . . 8 |- ( u = x -> ( ( j ↾t u ) e. A <-> ( j ↾t x ) e. A ) )
15	12, 14	anbi12d 747	. . . . . . 7 |- ( u = x -> ( ( y e. u /\ ( j ↾t u ) e. A ) <-> ( y e. x /\ ( j ↾t x ) e. A ) ) )
16	15	rspcev 3309	. . . . . 6 |- ( ( x e. ( j i^i ~P x ) /\ ( y e. x /\ ( j ↾t x ) e. A ) ) -> E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. A ) )
17	7, 8, 11, 16	syl12anc 1324	. . . . 5 |- ( ( ( ph /\ j e. A ) /\ ( x e. j /\ y e. x ) ) -> E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. A ) )
18	17	ralrimivva 2971	. . . 4 |- ( ( ph /\ j e. A ) -> A. x e. j A. y e. x E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. A ) )
19		islly 21271	. . . 4 |- ( j e. Locally A <-> ( j e. Top /\ A. x e. j A. y e. x E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. A ) ) )
20	2, 18, 19	sylanbrc 698	. . 3 |- ( ( ph /\ j e. A ) -> j e. Locally A )
21	20	ex 450	. 2 |- ( ph -> ( j e. A -> j e. Locally A ) )
22	21	ssrdv 3609	1 |- ( ph -> A C_ Locally A )

Syntax hints:   -> wi 4   /\ wa 384   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-lly 21269

This theorem is referenced by:  llyidm  21291  nllyidm  21292  toplly  21293  hauslly  21295  lly1stc  21299