Theorem imaundi

Description: Distributive law for image over union. Theorem 35 of Suppes p. 65. (Contributed by NM, 30-Sep-2002)

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

Proof

Step	Hyp	Ref	Expression
1		resundi	⊢ ( 𝐴 ↾ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐴 ↾ 𝐵 ) ∪ ( 𝐴 ↾ 𝐶 ) )
2	1	rneqi	⊢ ran ( 𝐴 ↾ ( 𝐵 ∪ 𝐶 ) ) = ran ( ( 𝐴 ↾ 𝐵 ) ∪ ( 𝐴 ↾ 𝐶 ) )
3		rnun	⊢ ran ( ( 𝐴 ↾ 𝐵 ) ∪ ( 𝐴 ↾ 𝐶 ) ) = ( ran ( 𝐴 ↾ 𝐵 ) ∪ ran ( 𝐴 ↾ 𝐶 ) )
4	2 3	eqtri	⊢ ran ( 𝐴 ↾ ( 𝐵 ∪ 𝐶 ) ) = ( ran ( 𝐴 ↾ 𝐵 ) ∪ ran ( 𝐴 ↾ 𝐶 ) )
5		df-ima	⊢ ( 𝐴 “ ( 𝐵 ∪ 𝐶 ) ) = ran ( 𝐴 ↾ ( 𝐵 ∪ 𝐶 ) )
6		df-ima	⊢ ( 𝐴 “ 𝐵 ) = ran ( 𝐴 ↾ 𝐵 )
7		df-ima	⊢ ( 𝐴 “ 𝐶 ) = ran ( 𝐴 ↾ 𝐶 )
8	6 7	uneq12i	⊢ ( ( 𝐴 “ 𝐵 ) ∪ ( 𝐴 “ 𝐶 ) ) = ( ran ( 𝐴 ↾ 𝐵 ) ∪ ran ( 𝐴 ↾ 𝐶 ) )
9	4 5 8	3eqtr4i	⊢ ( 𝐴 “ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐴 “ 𝐵 ) ∪ ( 𝐴 “ 𝐶 ) )