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Theorem imadifss 33016
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 3755 . . . . 5 𝐴 ⊆ (𝐵𝐴)
2 undif2 4016 . . . . 5 (𝐵 ∪ (𝐴𝐵)) = (𝐵𝐴)
31, 2sseqtr4i 3617 . . . 4 𝐴 ⊆ (𝐵 ∪ (𝐴𝐵))
4 imass2 5460 . . . 4 (𝐴 ⊆ (𝐵 ∪ (𝐴𝐵)) → (𝐹𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴𝐵))))
53, 4ax-mp 5 . . 3 (𝐹𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴𝐵)))
6 imaundi 5504 . . 3 (𝐹 “ (𝐵 ∪ (𝐴𝐵))) = ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵)))
75, 6sseqtri 3616 . 2 (𝐹𝐴) ⊆ ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵)))
8 ssundif 4024 . 2 ((𝐹𝐴) ⊆ ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵))) ↔ ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵)))
97, 8mpbi 220 1 ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  cdif 3552  cun 3553  wss 3555  cima 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087
This theorem is referenced by:  poimirlem30  33071
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