Theorem imaundir 5054

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 4651	. . 3 |- ( ( A u. B ) " C ) = ran ( ( A u. B ) |` C )
2		resundir 4933	. . . 4 |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )
3	2	rneqi 4867	. . 3 |- ran ( ( A u. B ) |` C ) = ran ( ( A |` C ) u. ( B |` C ) )
4		rnun 5049	. . 3 |- ran ( ( A |` C ) u. ( B |` C ) ) = ( ran ( A |` C ) u. ran ( B |` C ) )
5	1, 3, 4	3eqtri 2212	. 2 |- ( ( A u. B ) " C ) = ( ran ( A |` C ) u. ran ( B |` C ) )
6		df-ima 4651	. . 3 |- ( A " C ) = ran ( A |` C )
7		df-ima 4651	. . 3 |- ( B " C ) = ran ( B |` C )
8	6, 7	uneq12i 3299	. 2 |- ( ( A " C ) u. ( B " C ) ) = ( ran ( A |` C ) u. ran ( B |` C ) )
9	5, 8	eqtr4i 2211	1 |- ( ( A u. B ) " C ) = ( ( A " C ) u. ( B " C ) )

Syntax hints:   = wceq 1363   u. cun 3139   ran crn 4639   |` cres 4640   "cima 4641

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651

This theorem is referenced by:  fvun1  5595