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Theorem preddif 24262
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 3438 . 2  |-  ( ( A  \  B )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  \ 
( B  i^i  ( `' R " { X } ) ) )
2 df-pred 24239 . 2  |-  Pred ( R ,  ( A  \  B ) ,  X
)  =  ( ( A  \  B )  i^i  ( `' R " { X } ) )
3 df-pred 24239 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
4 df-pred 24239 . . 3  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
53, 4difeq12i 3305 . 2  |-  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  B ,  X ) )  =  ( ( A  i^i  ( `' R " { X } ) )  \ 
( B  i^i  ( `' R " { X } ) ) )
61, 2, 53eqtr4i 2326 1  |-  Pred ( R ,  ( A  \  B ) ,  X
)  =  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  B ,  X ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    \ cdif 3162    i^i cin 3164   {csn 3653   `'ccnv 4704   "cima 4708   Predcpred 24238
This theorem is referenced by:  wfrlem8  24334
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-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-in 3172  df-pred 24239
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