Theorem imadifss 37645

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

Assertion

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

Proof of Theorem imadifss

Step	Hyp	Ref	Expression
1		ssun2 4126	. . . . 5 ⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴)
2		undif2 4424	. . . . 5 ⊢ (𝐵 ∪ (𝐴 ∖ 𝐵)) = (𝐵 ∪ 𝐴)
3	1, 2	sseqtrri 3979	. . . 4 ⊢ 𝐴 ⊆ (𝐵 ∪ (𝐴 ∖ 𝐵))
4		imass2 6050	. . . 4 ⊢ (𝐴 ⊆ (𝐵 ∪ (𝐴 ∖ 𝐵)) → (𝐹 “ 𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴 ∖ 𝐵))))
5	3, 4	ax-mp 5	. . 3 ⊢ (𝐹 “ 𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴 ∖ 𝐵)))
6		imaundi 6096	. . 3 ⊢ (𝐹 “ (𝐵 ∪ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵)))
7	5, 6	sseqtri 3978	. 2 ⊢ (𝐹 “ 𝐴) ⊆ ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵)))
8		ssundif 4435	. 2 ⊢ ((𝐹 “ 𝐴) ⊆ ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵))) ↔ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵)))
9	7, 8	mpbi 230	1 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵))

Syntax hints:   ∖ cdif 3894   ∪ cun 3895   ⊆ wss 3897   “ cima 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627

This theorem is referenced by:  poimirlem30  37700