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Theorem trinxp 4812
Description: The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a square cross product is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.)
Assertion
Ref Expression
trinxp  |-  ( ( R  o.  R ) 
C_  R  ->  (
( R  i^i  ( A  X.  A ) )  o.  ( R  i^i  ( A  X.  A
) ) )  C_  ( R  i^i  ( A  X.  A ) ) )

Proof of Theorem trinxp
StepHypRef Expression
1 xpidtr 4809 . 2  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)
2 trin2 4810 . 2  |-  ( ( ( R  o.  R
)  C_  R  /\  ( ( A  X.  A )  o.  ( A  X.  A ) ) 
C_  ( A  X.  A ) )  -> 
( ( R  i^i  ( A  X.  A
) )  o.  ( R  i^i  ( A  X.  A ) ) ) 
C_  ( R  i^i  ( A  X.  A
) ) )
31, 2mpan2 416 1  |-  ( ( R  o.  R ) 
C_  R  ->  (
( R  i^i  ( A  X.  A ) )  o.  ( R  i^i  ( A  X.  A
) ) )  C_  ( R  i^i  ( A  X.  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    i^i cin 2996    C_ wss 2997    X. cxp 4426    o. ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-co 4437
This theorem is referenced by: (None)
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