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Theorem predidm 28833
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 28809 . 2  |-  Pred ( R ,  Pred ( R ,  A ,  X
) ,  X )  =  ( Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )
2 df-pred 28809 . . . . 5  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
3 inidm 3702 . . . . . 6  |-  ( ( `' R " { X } )  i^i  ( `' R " { X } ) )  =  ( `' R " { X } )
43ineq2i 3692 . . . . 5  |-  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )  =  ( A  i^i  ( `' R " { X } ) )
52, 4eqtr4i 2494 . . . 4  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )
6 inass 3703 . . . 4  |-  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )
75, 6eqtr4i 2494 . . 3  |-  Pred ( R ,  A ,  X )  =  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )
82ineq1i 3691 . . 3  |-  ( Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )
97, 8eqtr4i 2494 . 2  |-  Pred ( R ,  A ,  X )  =  (
Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )
101, 9eqtr4i 2494 1  |-  Pred ( R ,  Pred ( R ,  A ,  X
) ,  X )  =  Pred ( R ,  A ,  X )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    i^i cin 3470   {csn 4022   `'ccnv 4993   "cima 4997   Predcpred 28808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-v 3110  df-in 3478  df-pred 28809
This theorem is referenced by: (None)
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