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Theorem llyss 20162
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 3433 . . . . . . . 8  |-  ( A 
C_  B  ->  (
( jt  u )  e.  A  ->  ( jt  u )  e.  B
) )
21anim2d 563 . . . . . . 7  |-  ( A 
C_  B  ->  (
( y  e.  u  /\  ( jt  u )  e.  A
)  ->  ( y  e.  u  /\  (
jt  u )  e.  B
) ) )
32reximdv 2875 . . . . . 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 2811 . . . . 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 2811 . . . 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 563 . . 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 20151 . . 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 20151 . . 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 270 . 2  |-  ( A 
C_  B  ->  (
j  e. Locally  A  ->  j  e. Locally  B ) )
109ssrdv 3445 1  |-  ( A 
C_  B  -> Locally  A  C_ Locally  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1840   A.wral 2751   E.wrex 2752    i^i cin 3410    C_ wss 3411   ~Pcpw 3952  (class class class)co 6232   ↾t crest 14925   Topctop 19576  Locally clly 20147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-iota 5487  df-fv 5531  df-ov 6235  df-lly 20149
This theorem is referenced by: (None)
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