Theorem imaundi 5149

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 5026	. . . 4 |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )
2	1	rneqi 4960	. . 3 |- ran ( A |` ( B u. C ) ) = ran ( ( A |` B ) u. ( A |` C ) )
3		rnun 5145	. . 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 4738	. 2 |- ( A " ( B u. C ) ) = ran ( A |` ( B u. C ) )
6		df-ima 4738	. . 3 |- ( A " B ) = ran ( A |` B )
7		df-ima 4738	. . 3 |- ( A " C ) = ran ( A |` C )
8	6, 7	uneq12i 3359	. 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 1397   u. cun 3198   ran crn 4726   |` cres 4727   "cima 4728

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  fnimapr  5706  fiintim  7122  fidcenumlemrks  7151  fidcenumlemr  7153  resunimafz0  11094  ennnfonelemhf1o  13033