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Theorem nllyss 19059
Description: The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyss  |-  ( A 
C_  B  -> 𝑛Locally  A  C_ 𝑛Locally  B )

Proof of Theorem nllyss
Dummy variables  j  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3345 . . . . . . 7  |-  ( A 
C_  B  ->  (
( jt  u )  e.  A  ->  ( jt  u )  e.  B
) )
21reximdv 2822 . . . . . 6  |-  ( A 
C_  B  ->  ( E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A  ->  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  B ) )
32ralimdv 2790 . . . . 5  |-  ( A 
C_  B  ->  ( A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A  ->  A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  B ) )
43ralimdv 2790 . . . 4  |-  ( A 
C_  B  ->  ( A. x  e.  j  A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A  ->  A. x  e.  j  A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  B ) )
54anim2d 565 . . 3  |-  ( A 
C_  B  ->  (
( j  e.  Top  /\ 
A. x  e.  j 
A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A )  ->  ( j  e. 
Top  /\  A. x  e.  j  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  B
) ) )
6 isnlly 19048 . . 3  |-  ( j  e. 𝑛Locally  A  <->  ( j  e. 
Top  /\  A. x  e.  j  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
) )
7 isnlly 19048 . . 3  |-  ( j  e. 𝑛Locally  B  <->  ( j  e. 
Top  /\  A. x  e.  j  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  B
) )
85, 6, 73imtr4g 270 . 2  |-  ( A 
C_  B  ->  (
j  e. 𝑛Locally  A  ->  j  e. 𝑛Locally  B ) )
98ssrdv 3357 1  |-  ( A 
C_  B  -> 𝑛Locally  A  C_ 𝑛Locally  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   A.wral 2710   E.wrex 2711    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   {csn 3872   ` cfv 5413  (class class class)co 6086   ↾t crest 14351   Topctop 18473   neicnei 18676  𝑛Locally cnlly 19044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-iota 5376  df-fv 5421  df-ov 6089  df-nlly 19046
This theorem is referenced by:  iinllycon  27095  cvmlift3  27169
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