Theorem inres 4976

Description: Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)

Assertion

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

Proof of Theorem inres

Step	Hyp	Ref	Expression
1		inass 3383	. 2 |- ( ( A i^i B ) i^i ( C X. _V ) ) = ( A i^i ( B i^i ( C X. _V ) ) )
2		df-res 4687	. 2 |- ( ( A i^i B ) |` C ) = ( ( A i^i B ) i^i ( C X. _V ) )
3		df-res 4687	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	3	ineq2i 3371	. 2 |- ( A i^i ( B |` C ) ) = ( A i^i ( B i^i ( C X. _V ) ) )
5	1, 2, 4	3eqtr4ri 2237	1 |- ( A i^i ( B |` C ) ) = ( ( A i^i B ) |` C )

Syntax hints:   = wceq 1373   _Vcvv 2772   i^i cin 3165   X. cxp 4673   |` cres 4677

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-res 4687

This theorem is referenced by:  resindm  5001