Theorem imaun2 3461

Description: The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Assertion

Ref	Expression
imaun2	|- ((A u. B)"C) = ((A"C) u. (B"C))

Proof of Theorem imaun2

Step	Hyp	Ref	Expression
1		df-ima 3191	. . 3 |- ((A u. B)"C) = ran ((A u. B) |` C)
2		resundir 3379	. . . 4 |- ((A u. B) |` C) = ((A |` C) u. (B |` C))
3	2	rneqi 3340	. . 3 |- ran ((A u. B) |` C) = ran ((A |` C) u. (B |` C))
4		rnun 3457	. . 3 |- ran ((A |` C) u. (B |` C)) = (ran ( A |` C) u. ran ( B |` C))
5	1, 3, 4	3eqtr 1499	. 2 |- ((A u. B)"C) = (ran ( A |` C) u. ran ( B |` C))
6		df-ima 3191	. . 3 |- (A"C) = ran ( A |` C)
7		df-ima 3191	. . 3 |- (B"C) = ran ( B |` C)
8	6, 7	uneq12i 2182	. 2 |- ((A"C) u. (B"C)) = (ran ( A |` C) u. ran ( B |` C))
9	5, 8	eqtr4 1498	1 |- ((A u. B)"C) = ((A"C) u. (B"C))

Syntax hints:   = wceq 956   u. cun 2045  ran crn 3171   |` cres 3172  "cima 3173

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-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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