Theorem trinxp 5176

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 5173	. 2 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )
2		trin2 5174	. 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 429	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 3219   C_ wss 3220   X. cxp 4767   o. ccom 4773

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-co 4778

This theorem is referenced by: (None)