Theorem llyss 21282

Description: The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
llyss	|- ( A C_ B -> Locally A C_ Locally B )

Proof of Theorem llyss

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

Step	Hyp	Ref	Expression
1		ssel 3597	. . . . . . . 8 |- ( A C_ B -> ( ( j ↾t u ) e. A -> ( j ↾t u ) e. B ) )
2	1	anim2d 589	. . . . . . 7 |- ( A C_ B -> ( ( y e. u /\ ( j ↾t u ) e. A ) -> ( y e. u /\ ( j ↾t u ) e. B ) ) )
3	2	reximdv 3016	. . . . . 6 |- ( A C_ B -> ( E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. A ) -> E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. B ) ) )
4	3	ralimdv 2963	. . . . 5 |- ( A C_ B -> ( A. y e. x E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. A ) -> A. y e. x E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. B ) ) )
5	4	ralimdv 2963	. . . 4 |- ( A C_ B -> ( A. x e. j A. y e. x E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) 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. B ) ) )
6	5	anim2d 589	. . 3 |- ( A C_ B -> ( ( 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 ) ) -> ( 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. B ) ) ) )
7		islly 21271	. . 3 |- ( 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 ) ) )
8		islly 21271	. . 3 |- ( j e. Locally B <-> ( 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. B ) ) )
9	6, 7, 8	3imtr4g 285	. 2 |- ( A C_ B -> ( j e. Locally A -> j e. Locally B ) )
10	9	ssrdv 3609	1 |- ( A C_ B -> Locally A C_ Locally B )

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-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: (None)