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Theorem imadifss 31886
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  |-  ( ( F " A ) 
\  ( F " B ) )  C_  ( F " ( A 
\  B ) )

Proof of Theorem imadifss
StepHypRef Expression
1 ssun2 3631 . . . . 5  |-  A  C_  ( B  u.  A
)
2 undif2 3872 . . . . 5  |-  ( B  u.  ( A  \  B ) )  =  ( B  u.  A
)
31, 2sseqtr4i 3498 . . . 4  |-  A  C_  ( B  u.  ( A  \  B ) )
4 imass2 5221 . . . 4  |-  ( A 
C_  ( B  u.  ( A  \  B ) )  ->  ( F " A )  C_  ( F " ( B  u.  ( A  \  B ) ) ) )
53, 4ax-mp 5 . . 3  |-  ( F
" A )  C_  ( F " ( B  u.  ( A  \  B ) ) )
6 imaundi 5265 . . 3  |-  ( F
" ( B  u.  ( A  \  B ) ) )  =  ( ( F " B
)  u.  ( F
" ( A  \  B ) ) )
75, 6sseqtri 3497 . 2  |-  ( F
" A )  C_  ( ( F " B )  u.  ( F " ( A  \  B ) ) )
8 ssundif 3880 . 2  |-  ( ( F " A ) 
C_  ( ( F
" B )  u.  ( F " ( A  \  B ) ) )  <->  ( ( F
" A )  \ 
( F " B
) )  C_  ( F " ( A  \  B ) ) )
97, 8mpbi 212 1  |-  ( ( F " A ) 
\  ( F " B ) )  C_  ( F " ( A 
\  B ) )
Colors of variables: wff setvar class
Syntax hints:    \ cdif 3434    u. cun 3435    C_ wss 3437   "cima 4854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-xp 4857  df-cnv 4859  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864
This theorem is referenced by:  poimirlem30  31928
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