Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  predidm Unicode version

Theorem predidm 24259
Description: Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.)
Assertion
Ref Expression
predidm  |-  Pred ( R ,  Pred ( R ,  A ,  X
) ,  X )  =  Pred ( R ,  A ,  X )

Proof of Theorem predidm
StepHypRef Expression
1 df-pred 24239 . 2  |-  Pred ( R ,  Pred ( R ,  A ,  X
) ,  X )  =  ( Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )
2 df-pred 24239 . . . . 5  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
3 inidm 3391 . . . . . 6  |-  ( ( `' R " { X } )  i^i  ( `' R " { X } ) )  =  ( `' R " { X } )
43ineq2i 3380 . . . . 5  |-  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )  =  ( A  i^i  ( `' R " { X } ) )
52, 4eqtr4i 2319 . . . 4  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )
6 inass 3392 . . . 4  |-  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )
75, 6eqtr4i 2319 . . 3  |-  Pred ( R ,  A ,  X )  =  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )
82ineq1i 3379 . . 3  |-  ( Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )
97, 8eqtr4i 2319 . 2  |-  Pred ( R ,  A ,  X )  =  (
Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )
101, 9eqtr4i 2319 1  |-  Pred ( R ,  Pred ( R ,  A ,  X
) ,  X )  =  Pred ( R ,  A ,  X )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    i^i cin 3164   {csn 3653   `'ccnv 4704   "cima 4708   Predcpred 24238
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-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-pred 24239
  Copyright terms: Public domain W3C validator