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Theorem predidm 5391
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 5369 . 2  |-  Pred ( R ,  Pred ( R ,  A ,  X
) ,  X )  =  ( Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )
2 df-pred 5369 . . . . 5  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
3 inidm 3650 . . . . . 6  |-  ( ( `' R " { X } )  i^i  ( `' R " { X } ) )  =  ( `' R " { X } )
43ineq2i 3640 . . . . 5  |-  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )  =  ( A  i^i  ( `' R " { X } ) )
52, 4eqtr4i 2436 . . . 4  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )
6 inass 3651 . . . 4  |-  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )
75, 6eqtr4i 2436 . . 3  |-  Pred ( R ,  A ,  X )  =  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )
82ineq1i 3639 . . 3  |-  ( Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )
97, 8eqtr4i 2436 . 2  |-  Pred ( R ,  A ,  X )  =  (
Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )
101, 9eqtr4i 2436 1  |-  Pred ( R ,  Pred ( R ,  A ,  X
) ,  X )  =  Pred ( R ,  A ,  X )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1407    i^i cin 3415   {csn 3974   `'ccnv 4824   "cima 4828   Predcpred 5368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-v 3063  df-in 3423  df-pred 5369
This theorem is referenced by: (None)
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