Theorem resindi 4834

Description: Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)

Assertion

Ref	Expression
resindi	|- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) )

Proof of Theorem resindi

Step	Hyp	Ref	Expression
1		xpindir 4675	. . . 4 |- ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
2	1	ineq2i 3274	. . 3 |- ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
3		inindi 3293	. . 3 |- ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) ) = ( ( A i^i ( B X. _V ) ) i^i ( A i^i ( C X. _V ) ) )
4	2, 3	eqtri 2160	. 2 |- ( A i^i ( ( B i^i C ) X. _V ) ) = ( ( A i^i ( B X. _V ) ) i^i ( A i^i ( C X. _V ) ) )
5		df-res 4551	. 2 |- ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
6		df-res 4551	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res 4551	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6, 7	ineq12i 3275	. 2 |- ( ( A |` B ) i^i ( A |` C ) ) = ( ( A i^i ( B X. _V ) ) i^i ( A i^i ( C X. _V ) ) )
9	4, 5, 8	3eqtr4i 2170	1 |- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) )

Syntax hints:   = wceq 1331   _Vcvv 2686   i^i cin 3070   X. cxp 4537   |` cres 4541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545  df-rel 4546  df-res 4551

This theorem is referenced by:  resindm  4861