MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  llyeq Unicode version

Theorem llyeq 17454
Description: Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llyeq  |-  ( A  =  B  -> Locally  A  = Locally  B )

Proof of Theorem llyeq
Dummy variables  j  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2448 . . . . . 6  |-  ( A  =  B  ->  (
( jt  u )  e.  A  <->  ( jt  u )  e.  B
) )
21anbi2d 685 . . . . 5  |-  ( A  =  B  ->  (
( y  e.  u  /\  ( jt  u )  e.  A
)  <->  ( y  e.  u  /\  ( jt  u )  e.  B ) ) )
32rexbidv 2670 . . . 4  |-  ( A  =  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 ) ) )
432ralbidv 2691 . . 3  |-  ( A  =  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 ) ) )
54rabbidv 2891 . 2  |-  ( A  =  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 ) } )
6 df-lly 17450 . 2  |- 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 ) }
7 df-lly 17450 . 2  |- 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 ) }
85, 6, 73eqtr4g 2444 1  |-  ( A  =  B  -> Locally  A  = Locally  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650   {crab 2653    i^i cin 3262   ~Pcpw 3742  (class class class)co 6020   ↾t crest 13575   Topctop 16881  Locally clly 17448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-ral 2654  df-rex 2655  df-rab 2658  df-lly 17450
  Copyright terms: Public domain W3C validator