Theorem resindi 4924

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 4765	. . . 4 |- ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
2	1	ineq2i 3335	. . 3 |- ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
3		inindi 3354	. . 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 2198	. 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 4640	. 2 |- ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
6		df-res 4640	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res 4640	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6, 7	ineq12i 3336	. 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 2208	1 |- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) )

Syntax hints:   = wceq 1353   _Vcvv 2739   i^i cin 3130   X. cxp 4626   |` cres 4630

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-opab 4067  df-xp 4634  df-rel 4635  df-res 4640

This theorem is referenced by:  resindm  4951