Theorem resundi 3378

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 3226	. . . 4 |- ((B u. C) X. V) = ((B X. V) u. (C X. V))
2	1	ineq2i 2214	. . 3 |- (A i^i ((B u. C) X. V)) = (A i^i ((B X. V) u. (C X. V)))
3		indi 2251	. . 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 1495	. 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 3190	. 2 |- (A |` (B u. C)) = (A i^i ((B u. C) X. V))
6		df-res 3190	. . 3 |- (A |` B) = (A i^i (B X. V))
7		df-res 3190	. . 3 |- (A |` C) = (A i^i (C X. V))
8	6, 7	uneq12i 2182	. 2 |- ((A |` B) u. (A |` C)) = ((A i^i (B X. V)) u. (A i^i (C X. V)))
9	4, 5, 8	3eqtr4 1505	1 |- (A |` (B u. C)) = ((A |` B) u. (A |` C))

Syntax hints:   = wceq 956  Vcvv 1811   u. cun 2045   i^i cin 2046   X. cxp 3168   |` cres 3172

This theorem is referenced by:  imaun 3460  mapunen 4502

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-in 2051  df-opab 2667  df-xp 3184  df-res 3190

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