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Theorem llyss 20486
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.
StepHypRef Expression
1 ssel 3459 . . . . . . . 8  |-  ( A 
C_  B  ->  (
( jt  u )  e.  A  ->  ( jt  u )  e.  B
) )
21anim2d 568 . . . . . . 7  |-  ( A 
C_  B  ->  (
( y  e.  u  /\  ( jt  u )  e.  A
)  ->  ( y  e.  u  /\  (
jt  u )  e.  B
) ) )
32reximdv 2900 . . . . . 6  |-  ( A 
C_  B  ->  ( E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A )  ->  E. u  e.  ( j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  B ) ) )
43ralimdv 2836 . . . . 5  |-  ( A 
C_  B  ->  ( A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A )  ->  A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  B ) ) )
54ralimdv 2836 . . . 4  |-  ( A 
C_  B  ->  ( A. x  e.  j  A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A )  ->  A. x  e.  j 
A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  B ) ) )
65anim2d 568 . . 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  /\  ( jt  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  /\  (
jt  u )  e.  B
) ) ) )
7 islly 20475 . . 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  /\  (
jt  u )  e.  A
) ) )
8 islly 20475 . . 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  /\  (
jt  u )  e.  B
) ) )
96, 7, 83imtr4g 274 . 2  |-  ( A 
C_  B  ->  (
j  e. Locally  A  ->  j  e. Locally  B ) )
109ssrdv 3471 1  |-  ( A 
C_  B  -> Locally  A  C_ Locally  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1869   A.wral 2776   E.wrex 2777    i^i cin 3436    C_ wss 3437   ~Pcpw 3980  (class class class)co 6303   ↾t crest 15312   Topctop 19909  Locally clly 20471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-iota 5563  df-fv 5607  df-ov 6306  df-lly 20473
This theorem is referenced by: (None)
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