Theorem imaundir 4908

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 4510	. . 3 |- ( ( A u. B ) " C ) = ran ( ( A u. B ) |` C )
2		resundir 4789	. . . 4 |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )
3	2	rneqi 4725	. . 3 |- ran ( ( A u. B ) |` C ) = ran ( ( A |` C ) u. ( B |` C ) )
4		rnun 4903	. . 3 |- ran ( ( A |` C ) u. ( B |` C ) ) = ( ran ( A |` C ) u. ran ( B |` C ) )
5	1, 3, 4	3eqtri 2137	. 2 |- ( ( A u. B ) " C ) = ( ran ( A |` C ) u. ran ( B |` C ) )
6		df-ima 4510	. . 3 |- ( A " C ) = ran ( A |` C )
7		df-ima 4510	. . 3 |- ( B " C ) = ran ( B |` C )
8	6, 7	uneq12i 3192	. 2 |- ( ( A " C ) u. ( B " C ) ) = ( ran ( A |` C ) u. ran ( B |` C ) )
9	5, 8	eqtr4i 2136	1 |- ( ( A u. B ) " C ) = ( ( A " C ) u. ( B " C ) )

Syntax hints:   = wceq 1312   u. cun 3033   ran crn 4498   |` cres 4499   "cima 4500

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510

This theorem is referenced by:  fvun1  5439