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Theorem imadifss 37602
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 4179 . . . . 5 𝐴 ⊆ (𝐵𝐴)
2 undif2 4477 . . . . 5 (𝐵 ∪ (𝐴𝐵)) = (𝐵𝐴)
31, 2sseqtrri 4033 . . . 4 𝐴 ⊆ (𝐵 ∪ (𝐴𝐵))
4 imass2 6120 . . . 4 (𝐴 ⊆ (𝐵 ∪ (𝐴𝐵)) → (𝐹𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴𝐵))))
53, 4ax-mp 5 . . 3 (𝐹𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴𝐵)))
6 imaundi 6169 . . 3 (𝐹 “ (𝐵 ∪ (𝐴𝐵))) = ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵)))
75, 6sseqtri 4032 . 2 (𝐹𝐴) ⊆ ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵)))
8 ssundif 4488 . 2 ((𝐹𝐴) ⊆ ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵))) ↔ ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵)))
97, 8mpbi 230 1 ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  cdif 3948  cun 3949  wss 3951  cima 5688
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  poimirlem30  37657
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