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Theorem difres 28260
Description: Case when class difference in unaffected by restriction. (Contributed by Thierry Arnoux, 1-Jan-2020.)
Assertion
Ref Expression
difres  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( C  |`  B ) )  =  ( A  \  C
) )

Proof of Theorem difres
StepHypRef Expression
1 df-res 4865 . . 3  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
21difeq2i 3560 . 2  |-  ( A 
\  ( C  |`  B ) )  =  ( A  \  ( C  i^i  ( B  X.  _V ) ) )
3 difindi 3709 . . . 4  |-  ( A 
\  ( C  i^i  ( B  X.  _V )
) )  =  ( ( A  \  C
)  u.  ( A 
\  ( B  X.  _V ) ) )
4 ssdif 3580 . . . . . . 7  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( B  X.  _V ) )  C_  (
( B  X.  _V )  \  ( B  X.  _V ) ) )
5 difid 3847 . . . . . . 7  |-  ( ( B  X.  _V )  \  ( B  X.  _V ) )  =  (/)
64, 5syl6sseq 3490 . . . . . 6  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( B  X.  _V ) )  C_  (/) )
7 ss0 3777 . . . . . 6  |-  ( ( A  \  ( B  X.  _V ) ) 
C_  (/)  ->  ( A  \  ( B  X.  _V ) )  =  (/) )
86, 7syl 17 . . . . 5  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( B  X.  _V ) )  =  (/) )
98uneq2d 3600 . . . 4  |-  ( A 
C_  ( B  X.  _V )  ->  ( ( A  \  C )  u.  ( A  \ 
( B  X.  _V ) ) )  =  ( ( A  \  C )  u.  (/) ) )
103, 9syl5eq 2508 . . 3  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( C  i^i  ( B  X.  _V )
) )  =  ( ( A  \  C
)  u.  (/) ) )
11 un0 3771 . . 3  |-  ( ( A  \  C )  u.  (/) )  =  ( A  \  C )
1210, 11syl6eq 2512 . 2  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( C  i^i  ( B  X.  _V )
) )  =  ( A  \  C ) )
132, 12syl5eq 2508 1  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( C  |`  B ) )  =  ( A  \  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455   _Vcvv 3057    \ cdif 3413    u. cun 3414    i^i cin 3415    C_ wss 3416   (/)c0 3743    X. cxp 4851    |` cres 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-res 4865
This theorem is referenced by:  qtophaus  28712
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