Theorem imaundir 5079

Description: The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Assertion

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

Proof of Theorem imaundir

Step	Hyp	Ref	Expression
1		df-ima 4672	. . 3 |- ( ( A u. B ) " C ) = ran ( ( A u. B ) |` C )
2		resundir 4956	. . . 4 |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )
3	2	rneqi 4890	. . 3 |- ran ( ( A u. B ) |` C ) = ran ( ( A |` C ) u. ( B |` C ) )
4		rnun 5074	. . 3 |- ran ( ( A |` C ) u. ( B |` C ) ) = ( ran ( A |` C ) u. ran ( B |` C ) )
5	1, 3, 4	3eqtri 2218	. 2 |- ( ( A u. B ) " C ) = ( ran ( A |` C ) u. ran ( B |` C ) )
6		df-ima 4672	. . 3 |- ( A " C ) = ran ( A |` C )
7		df-ima 4672	. . 3 |- ( B " C ) = ran ( B |` C )
8	6, 7	uneq12i 3311	. 2 |- ( ( A " C ) u. ( B " C ) ) = ( ran ( A |` C ) u. ran ( B |` C ) )
9	5, 8	eqtr4i 2217	1 |- ( ( A u. B ) " C ) = ( ( A " C ) u. ( B " C ) )

Syntax hints:   = wceq 1364   u. cun 3151   ran crn 4660   |` cres 4661   "cima 4662

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 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672

This theorem is referenced by:  fvun1  5623