Theorem inimass

Description: The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017)

	Ref	Expression
Assertion	inimass	⊢ ( ( 𝐴 ∩ 𝐵 ) “ 𝐶 ) ⊆ ( ( 𝐴 “ 𝐶 ) ∩ ( 𝐵 “ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		rnin	⊢ ran ( ( 𝐴 ↾ 𝐶 ) ∩ ( 𝐵 ↾ 𝐶 ) ) ⊆ ( ran ( 𝐴 ↾ 𝐶 ) ∩ ran ( 𝐵 ↾ 𝐶 ) )
2		df-ima	⊢ ( ( 𝐴 ∩ 𝐵 ) “ 𝐶 ) = ran ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 )
3		resindir	⊢ ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ↾ 𝐶 ) ∩ ( 𝐵 ↾ 𝐶 ) )
4	3	rneqi	⊢ ran ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 ) = ran ( ( 𝐴 ↾ 𝐶 ) ∩ ( 𝐵 ↾ 𝐶 ) )
5	2 4	eqtri	⊢ ( ( 𝐴 ∩ 𝐵 ) “ 𝐶 ) = ran ( ( 𝐴 ↾ 𝐶 ) ∩ ( 𝐵 ↾ 𝐶 ) )
6		df-ima	⊢ ( 𝐴 “ 𝐶 ) = ran ( 𝐴 ↾ 𝐶 )
7		df-ima	⊢ ( 𝐵 “ 𝐶 ) = ran ( 𝐵 ↾ 𝐶 )
8	6 7	ineq12i	⊢ ( ( 𝐴 “ 𝐶 ) ∩ ( 𝐵 “ 𝐶 ) ) = ( ran ( 𝐴 ↾ 𝐶 ) ∩ ran ( 𝐵 ↾ 𝐶 ) )
9	1 5 8	3sstr4i	⊢ ( ( 𝐴 ∩ 𝐵 ) “ 𝐶 ) ⊆ ( ( 𝐴 “ 𝐶 ) ∩ ( 𝐵 “ 𝐶 ) )