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 33514
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 3810 . . . . 5 𝐴 ⊆ (𝐵𝐴)
2 undif2 4077 . . . . 5 (𝐵 ∪ (𝐴𝐵)) = (𝐵𝐴)
31, 2sseqtr4i 3671 . . . 4 𝐴 ⊆ (𝐵 ∪ (𝐴𝐵))
4 imass2 5536 . . . 4 (𝐴 ⊆ (𝐵 ∪ (𝐴𝐵)) → (𝐹𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴𝐵))))
53, 4ax-mp 5 . . 3 (𝐹𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴𝐵)))
6 imaundi 5580 . . 3 (𝐹 “ (𝐵 ∪ (𝐴𝐵))) = ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵)))
75, 6sseqtri 3670 . 2 (𝐹𝐴) ⊆ ((𝐹𝐵) ∪ (𝐹 “ (𝐴𝐵)))
8 ssundif 4085 . 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 3604  cun 3605  wss 3607  cima 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  poimirlem30  33569
  Copyright terms: Public domain W3C validator