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Theorem llyss 17221
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 3187 . . . . . . . 8  |-  ( A 
C_  B  ->  (
( jt  u )  e.  A  ->  ( jt  u )  e.  B
) )
21anim2d 548 . . . . . . 7  |-  ( A 
C_  B  ->  (
( y  e.  u  /\  ( jt  u )  e.  A
)  ->  ( y  e.  u  /\  (
jt  u )  e.  B
) ) )
32reximdv 2667 . . . . . 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 2635 . . . . 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 2635 . . . 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 548 . . 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 17210 . . 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 17210 . . 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 261 . 2  |-  ( A 
C_  B  ->  (
j  e. Locally  A  ->  j  e. Locally  B ) )
109ssrdv 3198 1  |-  ( A 
C_  B  -> Locally  A  C_ Locally  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   ~Pcpw 3638  (class class class)co 5874   ↾t crest 13341   Topctop 16647  Locally clly 17206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-lly 17208
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