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Theorem predun 24190
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 3417 . 2  |-  ( ( A  u.  B )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  u.  ( B  i^i  ( `' R " { X } ) ) )
2 df-pred 24168 . 2  |-  Pred ( R ,  ( A  u.  B ) ,  X
)  =  ( ( A  u.  B )  i^i  ( `' R " { X } ) )
3 df-pred 24168 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
4 df-pred 24168 . . 3  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
53, 4uneq12i 3327 . 2  |-  ( Pred ( R ,  A ,  X )  u.  Pred ( R ,  B ,  X ) )  =  ( ( A  i^i  ( `' R " { X } ) )  u.  ( B  i^i  ( `' R " { X } ) ) )
61, 2, 53eqtr4i 2313 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 1623    u. cun 3150    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-un 3157  df-in 3159  df-pred 24168
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