Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imadifss Structured version   Visualization version   GIF version

Theorem imadifss 36926
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 4173 . . . . 5 𝐴 ⊆ (𝐵𝐴)
2 undif2 4476 . . . . 5 (𝐵 ∪ (𝐴𝐵)) = (𝐵𝐴)
31, 2sseqtrri 4019 . . . 4 𝐴 ⊆ (𝐵 ∪ (𝐴𝐵))
4 imass2 6101 . . . 4 (𝐴 ⊆ (𝐵 ∪ (𝐴𝐵)) → (𝐹𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴𝐵))))
53, 4ax-mp 5 . . 3 (𝐹𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴𝐵)))
6 imaundi 6149 . . 3 (𝐹 “ (𝐵 ∪ (𝐴𝐵))) = ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵)))
75, 6sseqtri 4018 . 2 (𝐹𝐴) ⊆ ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵)))
8 ssundif 4487 . 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 3945  cun 3946  wss 3948  cima 5679
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  poimirlem30  36981
  Copyright terms: Public domain W3C validator