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Theorem predidm 24188
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 24168 . 2  |-  Pred ( R ,  Pred ( R ,  A ,  X
) ,  X )  =  ( Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )
2 df-pred 24168 . . . . 5  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
3 inidm 3378 . . . . . 6  |-  ( ( `' R " { X } )  i^i  ( `' R " { X } ) )  =  ( `' R " { X } )
43ineq2i 3367 . . . . 5  |-  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )  =  ( A  i^i  ( `' R " { X } ) )
52, 4eqtr4i 2306 . . . 4  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )
6 inass 3379 . . . 4  |-  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( ( `' R " { X } )  i^i  ( `' R " { X } ) ) )
75, 6eqtr4i 2306 . . 3  |-  Pred ( R ,  A ,  X )  =  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )
82ineq1i 3366 . . 3  |-  ( Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( `' R " { X } ) )
97, 8eqtr4i 2306 . 2  |-  Pred ( R ,  A ,  X )  =  (
Pred ( R ,  A ,  X )  i^i  ( `' R " { X } ) )
101, 9eqtr4i 2306 1  |-  Pred ( R ,  Pred ( R ,  A ,  X
) ,  X )  =  Pred ( R ,  A ,  X )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    i^i cin 3151   {csn 3640   `'ccnv 4688   "cima 4692   Predcpred 24167
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-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-pred 24168
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