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Theorem predun 25216
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 3534 . 2  |-  ( ( A  u.  B )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  u.  ( B  i^i  ( `' R " { X } ) ) )
2 df-pred 25194 . 2  |-  Pred ( R ,  ( A  u.  B ) ,  X
)  =  ( ( A  u.  B )  i^i  ( `' R " { X } ) )
3 df-pred 25194 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
4 df-pred 25194 . . 3  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
53, 4uneq12i 3444 . 2  |-  ( Pred ( R ,  A ,  X )  u.  Pred ( R ,  B ,  X ) )  =  ( ( A  i^i  ( `' R " { X } ) )  u.  ( B  i^i  ( `' R " { X } ) ) )
61, 2, 53eqtr4i 2419 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 3263    i^i cin 3264   {csn 3759   `'ccnv 4819   "cima 4823   Predcpred 25193
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
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 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-un 3270  df-in 3272  df-pred 25194
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