Theorem trinxp 5121

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

Step	Hyp	Ref	Expression
1		xpidtr 5118	. 2 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )
2		trin2 5119	. 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 ) ) )
3	1, 2	mpan2 425	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 ) ) )

Syntax hints:   -> wi 4   i^i cin 3196   C_ wss 3197   X. cxp 4716   o. ccom 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-co 4727

This theorem is referenced by: (None)