Theorem resundi 4982

Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by set.mm contributors, 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 4834	. . . 4 |- ((B u. C) X. _V) = ((B X. _V) u. (C X. _V))
2	1	ineq2i 3455	. . 3 |- (A i^i ((B u. C) X. _V)) = (A i^i ((B X. _V) u. (C X. _V)))
3		indi 3502	. . 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 2373	. 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 4789	. 2 |- (A |` (B u. C)) = (A i^i ((B u. C) X. _V))
6		df-res 4789	. . 3 |- (A |` B) = (A i^i (B X. _V))
7		df-res 4789	. . 3 |- (A |` C) = (A i^i (C X. _V))
8	6, 7	uneq12i 3417	. 2 |- ((A |` B) u. (A |` C)) = ((A i^i (B X. _V)) u. (A i^i (C X. _V)))
9	4, 5, 8	3eqtr4i 2383	1 |- (A |` (B u. C)) = ((A |` B) u. (A |` C))

Syntax hints:   = wceq 1642  _Vcvv 2860   u. cun 3208   i^i cin 3209   X. cxp 4771   |` cres 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-opab 4624  df-xp 4785  df-res 4789

This theorem is referenced by:  imaundi  5040