Theorem resindi 4899

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 4740	. . . 4 |- ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
2	1	ineq2i 3320	. . 3 |- ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
3		inindi 3339	. . 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 2186	. 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 4616	. 2 |- ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
6		df-res 4616	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res 4616	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6, 7	ineq12i 3321	. 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 2196	1 |- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` 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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-xp 4610  df-rel 4611  df-res 4616

This theorem is referenced by:  resindm  4926