Theorem imadifss

Description: The difference of images is a subset of the image of the difference. (Contributed by Brendan Leahy, 21-Aug-2020)

	Ref	Expression
Assertion	imadifss	⊢ ( ( 𝐹 “ 𝐴 ) ∖ ( 𝐹 “ 𝐵 ) ) ⊆ ( 𝐹 “ ( 𝐴 ∖ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ssun2	⊢ 𝐴 ⊆ ( 𝐵 ∪ 𝐴 )
2		undif2	⊢ ( 𝐵 ∪ ( 𝐴 ∖ 𝐵 ) ) = ( 𝐵 ∪ 𝐴 )
3	1 2	sseqtrri	⊢ 𝐴 ⊆ ( 𝐵 ∪ ( 𝐴 ∖ 𝐵 ) )
4		imass2	⊢ ( 𝐴 ⊆ ( 𝐵 ∪ ( 𝐴 ∖ 𝐵 ) ) → ( 𝐹 “ 𝐴 ) ⊆ ( 𝐹 “ ( 𝐵 ∪ ( 𝐴 ∖ 𝐵 ) ) ) )
5	3 4	ax-mp	⊢ ( 𝐹 “ 𝐴 ) ⊆ ( 𝐹 “ ( 𝐵 ∪ ( 𝐴 ∖ 𝐵 ) ) )
6		imaundi	⊢ ( 𝐹 “ ( 𝐵 ∪ ( 𝐴 ∖ 𝐵 ) ) ) = ( ( 𝐹 “ 𝐵 ) ∪ ( 𝐹 “ ( 𝐴 ∖ 𝐵 ) ) )
7	5 6	sseqtri	⊢ ( 𝐹 “ 𝐴 ) ⊆ ( ( 𝐹 “ 𝐵 ) ∪ ( 𝐹 “ ( 𝐴 ∖ 𝐵 ) ) )
8		ssundif	⊢ ( ( 𝐹 “ 𝐴 ) ⊆ ( ( 𝐹 “ 𝐵 ) ∪ ( 𝐹 “ ( 𝐴 ∖ 𝐵 ) ) ) ↔ ( ( 𝐹 “ 𝐴 ) ∖ ( 𝐹 “ 𝐵 ) ) ⊆ ( 𝐹 “ ( 𝐴 ∖ 𝐵 ) ) )
9	7 8	mpbi	⊢ ( ( 𝐹 “ 𝐴 ) ∖ ( 𝐹 “ 𝐵 ) ) ⊆ ( 𝐹 “ ( 𝐴 ∖ 𝐵 ) )