Theorem imaundir

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

	Ref	Expression
Assertion	imaundir	⊢ ( ( 𝐴 ∪ 𝐵 ) “ 𝐶 ) = ( ( 𝐴 “ 𝐶 ) ∪ ( 𝐵 “ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ima	⊢ ( ( 𝐴 ∪ 𝐵 ) “ 𝐶 ) = ran ( ( 𝐴 ∪ 𝐵 ) ↾ 𝐶 )
2		resundir	⊢ ( ( 𝐴 ∪ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ↾ 𝐶 ) ∪ ( 𝐵 ↾ 𝐶 ) )
3	2	rneqi	⊢ ran ( ( 𝐴 ∪ 𝐵 ) ↾ 𝐶 ) = ran ( ( 𝐴 ↾ 𝐶 ) ∪ ( 𝐵 ↾ 𝐶 ) )
4		rnun	⊢ ran ( ( 𝐴 ↾ 𝐶 ) ∪ ( 𝐵 ↾ 𝐶 ) ) = ( ran ( 𝐴 ↾ 𝐶 ) ∪ ran ( 𝐵 ↾ 𝐶 ) )
5	1 3 4	3eqtri	⊢ ( ( 𝐴 ∪ 𝐵 ) “ 𝐶 ) = ( ran ( 𝐴 ↾ 𝐶 ) ∪ 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 ) ∪ ( 𝐵 “ 𝐶 ) ) = ( ran ( 𝐴 ↾ 𝐶 ) ∪ ran ( 𝐵 ↾ 𝐶 ) )
9	5 8	eqtr4i	⊢ ( ( 𝐴 ∪ 𝐵 ) “ 𝐶 ) = ( ( 𝐴 “ 𝐶 ) ∪ ( 𝐵 “ 𝐶 ) )