Theorem imaun 3452

Description: Distributive law for image over union. Theorem 35 of [Suppes] p. 65.

Assertion

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

Proof of Theorem imaun

Step	Hyp	Ref	Expression
1		resundi 3370	. . . 4 |- (A |` (B u. C)) = ((A |` B) u. (A |` C))
2	1	rneqi 3335	. . 3 |- ran ( A |` (B u. C)) = ran ((A |` B) u. (A |` C))
3		rnun 3449	. . 3 |- ran ((A |` B) u. (A |` C)) = (ran ( A |` B) u. ran ( A |` C))
4	2, 3	eqtr 1492	. 2 |- ran ( A |` (B u. C)) = (ran ( A |` B) u. ran ( A |` C))
5		df-ima 3186	. 2 |- (A"(B u. C)) = ran ( A |` (B u. C))
6		df-ima 3186	. . 3 |- (A"B) = ran ( A |` B)
7		df-ima 3186	. . 3 |- (A"C) = ran ( A |` C)
8	6, 7	uneq12i 2178	. 2 |- ((A"B) u. (A"C)) = (ran ( A |` B) u. ran ( A |` C))
9	4, 5, 8	3eqtr4 1502	1 |- (A"(B u. C)) = ((A"B) u. (A"C))

Syntax hints:   = wceq 954   u. cun 2041  ran crn 3166   |` cres 3167  "cima 3168

This theorem is referenced by:  unifi 4538  fiint 4540  fodomfi 4546

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186

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