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Metamath Proof Explorer

Theorem imaundir

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

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

Proof

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