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Theorem predun 25404
Description: Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)
Assertion
Ref Expression
predun  |-  Pred ( R ,  ( A  u.  B ) ,  X
)  =  ( Pred ( R ,  A ,  X )  u.  Pred ( R ,  B ,  X ) )

Proof of Theorem predun
StepHypRef Expression
1 indir 3549 . 2  |-  ( ( A  u.  B )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  u.  ( B  i^i  ( `' R " { X } ) ) )
2 df-pred 25382 . 2  |-  Pred ( R ,  ( A  u.  B ) ,  X
)  =  ( ( A  u.  B )  i^i  ( `' R " { X } ) )
3 df-pred 25382 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
4 df-pred 25382 . . 3  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
53, 4uneq12i 3459 . 2  |-  ( Pred ( R ,  A ,  X )  u.  Pred ( R ,  B ,  X ) )  =  ( ( A  i^i  ( `' R " { X } ) )  u.  ( B  i^i  ( `' R " { X } ) ) )
61, 2, 53eqtr4i 2434 1  |-  Pred ( R ,  ( A  u.  B ) ,  X
)  =  ( Pred ( R ,  A ,  X )  u.  Pred ( R ,  B ,  X ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    u. cun 3278    i^i cin 3279   {csn 3774   `'ccnv 4836   "cima 4840   Predcpred 25381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-in 3287  df-pred 25382
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