Theorem imaundi 5114

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 4991	. . . 4 |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )
2	1	rneqi 4925	. . 3 |- ran ( A |` ( B u. C ) ) = ran ( ( A |` B ) u. ( A |` C ) )
3		rnun 5110	. . 3 |- ran ( ( A |` B ) u. ( A |` C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
4	2, 3	eqtri 2228	. 2 |- ran ( A |` ( B u. C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
5		df-ima 4706	. 2 |- ( A " ( B u. C ) ) = ran ( A |` ( B u. C ) )
6		df-ima 4706	. . 3 |- ( A " B ) = ran ( A |` B )
7		df-ima 4706	. . 3 |- ( A " C ) = ran ( A |` C )
8	6, 7	uneq12i 3333	. 2 |- ( ( A " B ) u. ( A " C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
9	4, 5, 8	3eqtr4i 2238	1 |- ( A " ( B u. C ) ) = ( ( A " B ) u. ( A " C ) )

Syntax hints:   = wceq 1373   u. cun 3172   ran crn 4694   |` cres 4695   "cima 4696

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by:  fnimapr  5662  fiintim  7054  fidcenumlemrks  7081  fidcenumlemr  7083  resunimafz0  11013  ennnfonelemhf1o  12899