Theorem resindir 4900

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 3340	. 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 4616	. 2 |- ( ( A i^i B ) |` C ) = ( ( A i^i B ) i^i ( C X. _V ) )
3		df-res 4616	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
4		df-res 4616	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
5	3, 4	ineq12i 3321	. 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 2196	1 |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )

Syntax hints:   = wceq 1343   _Vcvv 2726   i^i cin 3115   X. cxp 4602   |` cres 4606

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-res 4616

This theorem is referenced by:  inimass  5020  fnreseql  5595