Theorem resundir 3379

Description: Distributive law for restriction over union.

Assertion

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

Proof of Theorem resundir

Step	Hyp	Ref	Expression
1		indir 2253	. 2 |- ((A u. B) i^i (C X. V)) = ((A i^i (C X. V)) u. (B i^i (C X. V)))
2		df-res 3190	. 2 |- ((A u. B) |` C) = ((A u. B) i^i (C X. V))
3		df-res 3190	. . 3 |- (A |` C) = (A i^i (C X. V))
4		df-res 3190	. . 3 |- (B |` C) = (B i^i (C X. V))
5	3, 4	uneq12i 2182	. 2 |- ((A |` C) u. (B |` C)) = ((A i^i (C X. V)) u. (B i^i (C X. V)))
6	1, 2, 5	3eqtr4 1505	1 |- ((A u. B) |` C) = ((A |` C) u. (B |` C))

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

This theorem is referenced by:  imaun2 3461  fvsnun1 3795  fvsnun2 3796  mapunen 4502  acdc2lem2 7489  acdc5lem2 7492  ruclem6 7515

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-res 3190

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