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Theorem imadifss 35796
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 ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))

Proof of Theorem imadifss
StepHypRef Expression
1 ssun2 4113 . . . . 5 𝐴 ⊆ (𝐵𝐴)
2 undif2 4416 . . . . 5 (𝐵 ∪ (𝐴𝐵)) = (𝐵𝐴)
31, 2sseqtrri 3963 . . . 4 𝐴 ⊆ (𝐵 ∪ (𝐴𝐵))
4 imass2 6020 . . . 4 (𝐴 ⊆ (𝐵 ∪ (𝐴𝐵)) → (𝐹𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴𝐵))))
53, 4ax-mp 5 . . 3 (𝐹𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴𝐵)))
6 imaundi 6068 . . 3 (𝐹 “ (𝐵 ∪ (𝐴𝐵))) = ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵)))
75, 6sseqtri 3962 . 2 (𝐹𝐴) ⊆ ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵)))
8 ssundif 4424 . 2 ((𝐹𝐴) ⊆ ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵))) ↔ ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵)))
97, 8mpbi 229 1 ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  cdif 3889  cun 3890  wss 3892  cima 5603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-cnv 5608  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613
This theorem is referenced by:  poimirlem30  35851
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