Theorem imadifss 37796

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 4131	. . . . 5 ⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴)
2		undif2 4429	. . . . 5 ⊢ (𝐵 ∪ (𝐴 ∖ 𝐵)) = (𝐵 ∪ 𝐴)
3	1, 2	sseqtrri 3983	. . . 4 ⊢ 𝐴 ⊆ (𝐵 ∪ (𝐴 ∖ 𝐵))
4		imass2 6061	. . . 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 6107	. . 3 ⊢ (𝐹 “ (𝐵 ∪ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵)))
7	5, 6	sseqtri 3982	. 2 ⊢ (𝐹 “ 𝐴) ⊆ ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵)))
8		ssundif 4440	. 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 3898   ∪ cun 3899   ⊆ wss 3901   “ cima 5627

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 2115  ax-9 2123  ax-ext 2708

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-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  poimirlem30  37851