Theorem resindi 4957

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 4798	. . . 4 |- ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
2	1	ineq2i 3357	. . 3 |- ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
3		inindi 3376	. . 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 2214	. 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 4671	. 2 |- ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
6		df-res 4671	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res 4671	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6, 7	ineq12i 3358	. 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 2224	1 |- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) )

Syntax hints:   = wceq 1364   _Vcvv 2760   i^i cin 3152   X. cxp 4657   |` cres 4661

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091  df-xp 4665  df-rel 4666  df-res 4671

This theorem is referenced by:  resindm  4984