Theorem imaundi 5066

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 4945	. . . 4 |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )
2	1	rneqi 4880	. . 3 |- ran ( A |` ( B u. C ) ) = ran ( ( A |` B ) u. ( A |` C ) )
3		rnun 5062	. . 3 |- ran ( ( A |` B ) u. ( A |` C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
4	2, 3	eqtri 2210	. 2 |- ran ( A |` ( B u. C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
5		df-ima 4664	. 2 |- ( A " ( B u. C ) ) = ran ( A |` ( B u. C ) )
6		df-ima 4664	. . 3 |- ( A " B ) = ran ( A |` B )
7		df-ima 4664	. . 3 |- ( A " C ) = ran ( A |` C )
8	6, 7	uneq12i 3306	. 2 |- ( ( A " B ) u. ( A " C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
9	4, 5, 8	3eqtr4i 2220	1 |- ( A " ( B u. C ) ) = ( ( A " B ) u. ( A " C ) )

Syntax hints:   = wceq 1364   u. cun 3146   ran crn 4652   |` cres 4653   "cima 4654

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2758  df-un 3152  df-in 3154  df-ss 3161  df-sn 3620  df-pr 3621  df-op 3623  df-br 4026  df-opab 4087  df-xp 4657  df-cnv 4659  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664

This theorem is referenced by:  fnimapr  5604  fiintim  6965  fidcenumlemrks  6991  fidcenumlemr  6993  resunimafz0  10858  ennnfonelemhf1o  12481