Theorem resundi 4643

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 4415	. . . 4 |- ( ( B u. C ) X. _V ) = ( ( B X. _V ) u. ( C X. _V ) )
2	1	ineq2i 3164	. . 3 |- ( A i^i ( ( B u. C ) X. _V ) ) = ( A i^i ( ( B X. _V ) u. ( C X. _V ) ) )
3		indi 3211	. . 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 2101	. 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 4375	. 2 |- ( A |` ( B u. C ) ) = ( A i^i ( ( B u. C ) X. _V ) )
6		df-res 4375	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res 4375	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6, 7	uneq12i 3124	. 2 |- ( ( A |` B ) u. ( A |` C ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
9	4, 5, 8	3eqtr4i 2111	1 |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )

Syntax hints:   = wceq 1284   _Vcvv 2601   u. cun 2971   i^i cin 2972   X. cxp 4361   |` cres 4365

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-opab 3840  df-xp 4369  df-res 4375

This theorem is referenced by:  imaundi  4756  relresfld  4867  relcoi1  4869  resasplitss  5089  fnfi  6388  fseq1p1m1  9111