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Theorem difres 28201
Description: Case when set 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 4862 . . 3  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
21difeq2i 3580 . 2  |-  ( A 
\  ( C  |`  B ) )  =  ( A  \  ( C  i^i  ( B  X.  _V ) ) )
3 difindi 3727 . . . 4  |-  ( A 
\  ( C  i^i  ( B  X.  _V )
) )  =  ( ( A  \  C
)  u.  ( A 
\  ( B  X.  _V ) ) )
4 ssdif 3600 . . . . . . 7  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( B  X.  _V ) )  C_  (
( B  X.  _V )  \  ( B  X.  _V ) ) )
5 difid 3863 . . . . . . 7  |-  ( ( B  X.  _V )  \  ( B  X.  _V ) )  =  (/)
64, 5syl6sseq 3510 . . . . . 6  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( B  X.  _V ) )  C_  (/) )
7 ss0 3793 . . . . . 6  |-  ( ( A  \  ( B  X.  _V ) ) 
C_  (/)  ->  ( A  \  ( B  X.  _V ) )  =  (/) )
86, 7syl 17 . . . . 5  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( B  X.  _V ) )  =  (/) )
98uneq2d 3620 . . . 4  |-  ( A 
C_  ( B  X.  _V )  ->  ( ( A  \  C )  u.  ( A  \ 
( B  X.  _V ) ) )  =  ( ( A  \  C )  u.  (/) ) )
103, 9syl5eq 2475 . . 3  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( C  i^i  ( B  X.  _V )
) )  =  ( ( A  \  C
)  u.  (/) ) )
11 un0 3787 . . 3  |-  ( ( A  \  C )  u.  (/) )  =  ( A  \  C )
1210, 11syl6eq 2479 . 2  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( C  i^i  ( B  X.  _V )
) )  =  ( A  \  C ) )
132, 12syl5eq 2475 1  |-  ( A 
C_  ( B  X.  _V )  ->  ( A 
\  ( C  |`  B ) )  =  ( A  \  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   _Vcvv 3081    \ cdif 3433    u. cun 3434    i^i cin 3435    C_ wss 3436   (/)c0 3761    X. cxp 4848    |` cres 4852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-res 4862
This theorem is referenced by:  qtophaus  28659
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