Theorem imaundi 5156

Description: Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Assertion

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

Proof of Theorem imaundi

Step	Hyp	Ref	Expression
1		resundi 5032	. . . 4 |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )
2	1	rneqi 4966	. . 3 |- ran ( A |` ( B u. C ) ) = ran ( ( A |` B ) u. ( A |` C ) )
3		rnun 5152	. . 3 |- ran ( ( A |` B ) u. ( A |` C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
4	2, 3	eqtri 2252	. 2 |- ran ( A |` ( B u. C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
5		df-ima 4744	. 2 |- ( A " ( B u. C ) ) = ran ( A |` ( B u. C ) )
6		df-ima 4744	. . 3 |- ( A " B ) = ran ( A |` B )
7		df-ima 4744	. . 3 |- ( A " C ) = ran ( A |` C )
8	6, 7	uneq12i 3361	. 2 |- ( ( A " B ) u. ( A " C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
9	4, 5, 8	3eqtr4i 2262	1 |- ( A " ( B u. C ) ) = ( ( A " B ) u. ( A " C ) )

Syntax hints:   = wceq 1398   u. cun 3199   ran crn 4732   |` cres 4733   "cima 4734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  fnimapr  5715  fiintim  7166  fidcenumlemrks  7195  fidcenumlemr  7197  resunimafz0  11158  ennnfonelemhf1o  13114