Theorem imadifss 38134

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 4140	. . . . 5 ⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴)
2		undif2 4443	. . . . 5 ⊢ (𝐵 ∪ (𝐴 ∖ 𝐵)) = (𝐵 ∪ 𝐴)
3	1, 2	sseqtrri 3994	. . . 4 ⊢ 𝐴 ⊆ (𝐵 ∪ (𝐴 ∖ 𝐵))
4		imass2 6105	. . . 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 6148	. . 3 ⊢ (𝐹 “ (𝐵 ∪ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵)))
7	5, 6	sseqtri 3993	. 2 ⊢ (𝐹 “ 𝐴) ⊆ ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵)))
8		ssundif 4453	. 2 ⊢ ((𝐹 “ 𝐴) ⊆ ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵))) ↔ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵)))
9	7, 8	mpbi 233	1 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵))

Syntax hints:   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913   “ cima 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675

This theorem is referenced by:  poimirlem30  38189