Metamath Proof Explorer


Theorem difres

Description: Case when class difference in unaffected by restriction. (Contributed by Thierry Arnoux, 1-Jan-2020)

Ref Expression
Assertion difres
|- ( A C_ ( B X. _V ) -> ( A \ ( C |` B ) ) = ( A \ C ) )

Proof

Step Hyp Ref Expression
1 df-res
 |-  ( C |` B ) = ( C i^i ( B X. _V ) )
2 1 difeq2i
 |-  ( A \ ( C |` B ) ) = ( A \ ( C i^i ( B X. _V ) ) )
3 difindi
 |-  ( A \ ( C i^i ( B X. _V ) ) ) = ( ( A \ C ) u. ( A \ ( B X. _V ) ) )
4 ssdif
 |-  ( A C_ ( B X. _V ) -> ( A \ ( B X. _V ) ) C_ ( ( B X. _V ) \ ( B X. _V ) ) )
5 difid
 |-  ( ( B X. _V ) \ ( B X. _V ) ) = (/)
6 4 5 sseqtrdi
 |-  ( A C_ ( B X. _V ) -> ( A \ ( B X. _V ) ) C_ (/) )
7 ss0
 |-  ( ( A \ ( B X. _V ) ) C_ (/) -> ( A \ ( B X. _V ) ) = (/) )
8 6 7 syl
 |-  ( A C_ ( B X. _V ) -> ( A \ ( B X. _V ) ) = (/) )
9 8 uneq2d
 |-  ( A C_ ( B X. _V ) -> ( ( A \ C ) u. ( A \ ( B X. _V ) ) ) = ( ( A \ C ) u. (/) ) )
10 3 9 syl5eq
 |-  ( A C_ ( B X. _V ) -> ( A \ ( C i^i ( B X. _V ) ) ) = ( ( A \ C ) u. (/) ) )
11 un0
 |-  ( ( A \ C ) u. (/) ) = ( A \ C )
12 10 11 eqtrdi
 |-  ( A C_ ( B X. _V ) -> ( A \ ( C i^i ( B X. _V ) ) ) = ( A \ C ) )
13 2 12 syl5eq
 |-  ( A C_ ( B X. _V ) -> ( A \ ( C |` B ) ) = ( A \ C ) )