Theorem imaundir 5196

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 4782	. . 3 |- ( ( A u. B ) " C ) = ran ( ( A u. B ) |` C )
2		resundir 5072	. . . 4 |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )
3	2	rneqi 5005	. . 3 |- ran ( ( A u. B ) |` C ) = ran ( ( A |` C ) u. ( B |` C ) )
4		rnun 5191	. . 3 |- ran ( ( A |` C ) u. ( B |` C ) ) = ( ran ( A |` C ) u. ran ( B |` C ) )
5	1, 3, 4	3eqtri 2263	. 2 |- ( ( A u. B ) " C ) = ( ran ( A |` C ) u. ran ( B |` C ) )
6		df-ima 4782	. . 3 |- ( A " C ) = ran ( A |` C )
7		df-ima 4782	. . 3 |- ( B " C ) = ran ( B |` C )
8	6, 7	uneq12i 3381	. 2 |- ( ( A " C ) u. ( B " C ) ) = ( ran ( A |` C ) u. ran ( B |` C ) )
9	5, 8	eqtr4i 2262	1 |- ( ( A u. B ) " C ) = ( ( A " C ) u. ( B " C ) )

Syntax hints:   = wceq 1402   u. cun 3218   ran crn 4770   |` cres 4771   "cima 4772

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782

This theorem is referenced by:  fvun1  5763