Theorem inpreima

Description: Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Mario Carneiro, 14-Jun-2016)

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

Proof

Step	Hyp	Ref	Expression
1		funcnvcnv	⊢ ( Fun 𝐹 → Fun ◡ ◡ 𝐹 )
2		imain	⊢ ( Fun ◡ ◡ 𝐹 → ( ◡ 𝐹 “ ( 𝐴 ∩ 𝐵 ) ) = ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ 𝐵 ) ) )
3	1 2	syl	⊢ ( Fun 𝐹 → ( ◡ 𝐹 “ ( 𝐴 ∩ 𝐵 ) ) = ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ 𝐵 ) ) )