Theorem resindir 4835

Description: Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)

Assertion

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

Proof of Theorem resindir

Step	Hyp	Ref	Expression
1		inindir 3294	. 2 |- ( ( A i^i B ) i^i ( C X. _V ) ) = ( ( A i^i ( C X. _V ) ) i^i ( B i^i ( C X. _V ) ) )
2		df-res 4551	. 2 |- ( ( A i^i B ) |` C ) = ( ( A i^i B ) i^i ( C X. _V ) )
3		df-res 4551	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
4		df-res 4551	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
5	3, 4	ineq12i 3275	. 2 |- ( ( A |` C ) i^i ( B |` C ) ) = ( ( A i^i ( C X. _V ) ) i^i ( B i^i ( C X. _V ) ) )
6	1, 2, 5	3eqtr4i 2170	1 |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` 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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-res 4551

This theorem is referenced by:  inimass  4955  fnreseql  5530