Theorem imaun 2647

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 2582	. . . 4 |- (A |` (B u. C)) = ((A |` B) u. (A |` C))
2	1	rneqi 2556	. . 3 |- ran (A |` (B u. C)) = ran ((A |` B) u. (A |` C))
3		rnun 2644	. . 3 |- ran ((A |` B) u. (A |` C)) = (ran (A |` B) u. ran (A |` C))
4	2, 3	eqtr 1119	. 2 |- ran (A |` (B u. C)) = (ran (A |` B) u. ran (A |` C))
5		df-ima 2431	. 2 |- (A"(B u. C)) = ran (A |` (B u. C))
6		df-ima 2431	. . 3 |- (A"B) = ran (A |` B)
7		df-ima 2431	. . 3 |- (A"C) = ran (A |` C)
8	6, 7	uneq12i 1609	. 2 |- ((A"B) u. (A"C)) = (ran (A |` B) u. ran (A |` C))
9	4, 5, 8	3eqtr4 1126	1 |- (A"(B u. C)) = ((A"B) u. (A"C))

Syntax hints:   = wceq 1091   u. cun 1485  ran crn 2411   |` cres 2412  "cima 2413

This theorem is referenced by:  fiint 3445

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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