Theorem resundi 4904

Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
resundi	|- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )

Proof of Theorem resundi

Step	Hyp	Ref	Expression
1		xpundir 4668	. . . 4 |- ( ( B u. C ) X. _V ) = ( ( B X. _V ) u. ( C X. _V ) )
2	1	ineq2i 3325	. . 3 |- ( A i^i ( ( B u. C ) X. _V ) ) = ( A i^i ( ( B X. _V ) u. ( C X. _V ) ) )
3		indi 3374	. . 3 |- ( A i^i ( ( B X. _V ) u. ( C X. _V ) ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
4	2, 3	eqtri 2191	. 2 |- ( A i^i ( ( B u. C ) X. _V ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
5		df-res 4623	. 2 |- ( A |` ( B u. C ) ) = ( A i^i ( ( B u. C ) X. _V ) )
6		df-res 4623	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res 4623	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6, 7	uneq12i 3279	. 2 |- ( ( A |` B ) u. ( A |` C ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
9	4, 5, 8	3eqtr4i 2201	1 |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )

Syntax hints:   = wceq 1348   _Vcvv 2730   u. cun 3119   i^i cin 3120   X. cxp 4609   |` cres 4613

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-opab 4051  df-xp 4617  df-res 4623

This theorem is referenced by:  imaundi  5023  relresfld  5140  relcoi1  5142  resasplitss  5377  fnsnsplitss  5695  fnsnsplitdc  6484  fnfi  6914  fseq1p1m1  10050  resunimafz0  10766