Theorem resindir 5027

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 3423	. 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 4735	. 2 |- ( ( A i^i B ) |` C ) = ( ( A i^i B ) i^i ( C X. _V ) )
3		df-res 4735	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
4		df-res 4735	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
5	3, 4	ineq12i 3404	. 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 2260	1 |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )

Syntax hints:   = wceq 1395   _Vcvv 2800   i^i cin 3197   X. cxp 4721   |` cres 4725

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-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-in 3204  df-res 4735

This theorem is referenced by:  inimass  5151  fnreseql  5753