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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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