Theorem imain

Description: The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009)

	Ref	Expression
Assertion	imain	⊢ ( Fun ◡ 𝐹 → ( 𝐹 “ ( 𝐴 ∩ 𝐵 ) ) = ( ( 𝐹 “ 𝐴 ) ∩ ( 𝐹 “ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imadif	⊢ ( Fun ◡ 𝐹 → ( 𝐹 “ ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) ) ) = ( ( 𝐹 “ 𝐴 ) ∖ ( 𝐹 “ ( 𝐴 ∖ 𝐵 ) ) ) )
2		imadif	⊢ ( Fun ◡ 𝐹 → ( 𝐹 “ ( 𝐴 ∖ 𝐵 ) ) = ( ( 𝐹 “ 𝐴 ) ∖ ( 𝐹 “ 𝐵 ) ) )
3	2	difeq2d	⊢ ( Fun ◡ 𝐹 → ( ( 𝐹 “ 𝐴 ) ∖ ( 𝐹 “ ( 𝐴 ∖ 𝐵 ) ) ) = ( ( 𝐹 “ 𝐴 ) ∖ ( ( 𝐹 “ 𝐴 ) ∖ ( 𝐹 “ 𝐵 ) ) ) )
4	1 3	eqtrd	⊢ ( Fun ◡ 𝐹 → ( 𝐹 “ ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) ) ) = ( ( 𝐹 “ 𝐴 ) ∖ ( ( 𝐹 “ 𝐴 ) ∖ ( 𝐹 “ 𝐵 ) ) ) )
5		dfin4	⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) )
6	5	imaeq2i	⊢ ( 𝐹 “ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐹 “ ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) ) )
7		dfin4	⊢ ( ( 𝐹 “ 𝐴 ) ∩ ( 𝐹 “ 𝐵 ) ) = ( ( 𝐹 “ 𝐴 ) ∖ ( ( 𝐹 “ 𝐴 ) ∖ ( 𝐹 “ 𝐵 ) ) )
8	4 6 7	3eqtr4g	⊢ ( Fun ◡ 𝐹 → ( 𝐹 “ ( 𝐴 ∩ 𝐵 ) ) = ( ( 𝐹 “ 𝐴 ) ∩ ( 𝐹 “ 𝐵 ) ) )