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Theorem predin 25456
Description: Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)
Assertion
Ref Expression
predin  |-  Pred ( R ,  ( A  i^i  B ) ,  X
)  =  ( Pred ( R ,  A ,  X )  i^i  Pred ( R ,  B ,  X ) )

Proof of Theorem predin
StepHypRef Expression
1 inindir 3551 . 2  |-  ( ( A  i^i  B )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( B  i^i  ( `' R " { X } ) ) )
2 df-pred 25431 . 2  |-  Pred ( R ,  ( A  i^i  B ) ,  X
)  =  ( ( A  i^i  B )  i^i  ( `' R " { X } ) )
3 df-pred 25431 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
4 df-pred 25431 . . 3  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
53, 4ineq12i 3532 . 2  |-  ( Pred ( R ,  A ,  X )  i^i  Pred ( R ,  B ,  X ) )  =  ( ( A  i^i  ( `' R " { X } ) )  i^i  ( B  i^i  ( `' R " { X } ) ) )
61, 2, 53eqtr4i 2465 1  |-  Pred ( R ,  ( A  i^i  B ) ,  X
)  =  ( Pred ( R ,  A ,  X )  i^i  Pred ( R ,  B ,  X ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    i^i cin 3311   {csn 3806   `'ccnv 4869   "cima 4873   Predcpred 25430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-pred 25431
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