Theorem imaundir 4992

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 4592	. . 3 |- ( ( A u. B ) " C ) = ran ( ( A u. B ) |` C )
2		resundir 4873	. . . 4 |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )
3	2	rneqi 4807	. . 3 |- ran ( ( A u. B ) |` C ) = ran ( ( A |` C ) u. ( B |` C ) )
4		rnun 4987	. . 3 |- ran ( ( A |` C ) u. ( B |` C ) ) = ( ran ( A |` C ) u. ran ( B |` C ) )
5	1, 3, 4	3eqtri 2179	. 2 |- ( ( A u. B ) " C ) = ( ran ( A |` C ) u. ran ( B |` C ) )
6		df-ima 4592	. . 3 |- ( A " C ) = ran ( A |` C )
7		df-ima 4592	. . 3 |- ( B " C ) = ran ( B |` C )
8	6, 7	uneq12i 3255	. 2 |- ( ( A " C ) u. ( B " C ) ) = ( ran ( A |` C ) u. ran ( B |` C ) )
9	5, 8	eqtr4i 2178	1 |- ( ( A u. B ) " C ) = ( ( A " C ) u. ( B " C ) )

Syntax hints:   = wceq 1332   u. cun 3096   ran crn 4580   |` cres 4581   "cima 4582

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-cnv 4587  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592

This theorem is referenced by:  fvun1  5527