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Theorem preddif 25468
Description: Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
preddif  |-  Pred ( R ,  ( A  \  B ) ,  X
)  =  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  B ,  X ) )

Proof of Theorem preddif
StepHypRef Expression
1 indifdir 3599 . 2  |-  ( ( A  \  B )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  \ 
( B  i^i  ( `' R " { X } ) ) )
2 df-pred 25441 . 2  |-  Pred ( R ,  ( A  \  B ) ,  X
)  =  ( ( A  \  B )  i^i  ( `' R " { X } ) )
3 df-pred 25441 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
4 df-pred 25441 . . 3  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
53, 4difeq12i 3465 . 2  |-  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  B ,  X ) )  =  ( ( A  i^i  ( `' R " { X } ) )  \ 
( B  i^i  ( `' R " { X } ) ) )
61, 2, 53eqtr4i 2468 1  |-  Pred ( R ,  ( A  \  B ) ,  X
)  =  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  B ,  X ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    \ cdif 3319    i^i cin 3321   {csn 3816   `'ccnv 4879   "cima 4883   Predcpred 25440
This theorem is referenced by:  wfrlem8  25547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-in 3329  df-pred 25441
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