Theorem resundi 3384

Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65.

Assertion

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

Proof of Theorem resundi

Step	Hyp	Ref	Expression
1		xpundir 3232	. . . 4 |- ((B u. C) X. V) = ((B X. V) u. (C X. V))
2	1	ineq2i 2217	. . 3 |- (A i^i ((B u. C) X. V)) = (A i^i ((B X. V) u. (C X. V)))
3		indi 2254	. . 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	eqtr 1498	. 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 3196	. 2 |- (A |` (B u. C)) = (A i^i ((B u. C) X. V))
6		df-res 3196	. . 3 |- (A |` B) = (A i^i (B X. V))
7		df-res 3196	. . 3 |- (A |` C) = (A i^i (C X. V))
8	6, 7	uneq12i 2185	. 2 |- ((A |` B) u. (A |` C)) = ((A i^i (B X. V)) u. (A i^i (C X. V)))
9	4, 5, 8	3eqtr4 1508	1 |- (A |` (B u. C)) = ((A |` B) u. (A |` C))

Syntax hints:   = wceq 958  Vcvv 1814   u. cun 2048   i^i cin 2049   X. cxp 3174   |` cres 3178

This theorem is referenced by:  imaun 3466  mapunen 4508

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054  df-opab 2672  df-xp 3190  df-res 3196

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