Theorem imaundi 5023

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 4904	. . . 4 |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )
2	1	rneqi 4839	. . 3 |- ran ( A |` ( B u. C ) ) = ran ( ( A |` B ) u. ( A |` C ) )
3		rnun 5019	. . 3 |- ran ( ( A |` B ) u. ( A |` C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
4	2, 3	eqtri 2191	. 2 |- ran ( A |` ( B u. C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
5		df-ima 4624	. 2 |- ( A " ( B u. C ) ) = ran ( A |` ( B u. C ) )
6		df-ima 4624	. . 3 |- ( A " B ) = ran ( A |` B )
7		df-ima 4624	. . 3 |- ( A " C ) = ran ( A |` C )
8	6, 7	uneq12i 3279	. 2 |- ( ( A " B ) u. ( A " C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) )
9	4, 5, 8	3eqtr4i 2201	1 |- ( A " ( B u. C ) ) = ( ( A " B ) u. ( A " C ) )

Syntax hints:   = wceq 1348   u. cun 3119   ran crn 4612   |` cres 4613   "cima 4614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by:  fnimapr  5556  fiintim  6906  fidcenumlemrks  6930  fidcenumlemr  6932  resunimafz0  10766  ennnfonelemhf1o  12368