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Theorem nllyss 17206
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 3174 . . . . . . 7  |-  ( A 
C_  B  ->  (
( jt  u )  e.  A  ->  ( jt  u )  e.  B
) )
21reximdv 2654 . . . . . 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 2622 . . . . 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 2622 . . . 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 548 . . 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 17195 . . 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 17195 . . 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 261 . 2  |-  ( A 
C_  B  ->  (
j  e. 𝑛Locally  A  ->  j  e. 𝑛Locally  B ) )
98ssrdv 3185 1  |-  ( A 
C_  B  -> 𝑛Locally  A  C_ 𝑛Locally  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631   neicnei 16834  𝑛Locally cnlly 17191
This theorem is referenced by:  iinllycon  23785  cvmlift3  23859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-nlly 17193
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