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Theorem difres 29413
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 5126 . . 3 |- ( C |` B ) = ( C i^i ( B X. _V ) )
21difeq2i 3725 . 2 |- ( A \ ( C |` B ) ) = ( A \ ( C i^i ( B X. _V ) ) )
3 difindi 3881 . . . 4 |- ( A \ ( C i^i ( B X. _V ) ) ) = ( ( A \ C ) u. ( A \ ( B X. _V ) ) )
4 ssdif 3745 . . . . . . 7 |- ( A C_ ( B X. _V ) -> ( A \ ( B X. _V ) ) C_ ( ( B X. _V ) \ ( B X. _V ) ) )
5 difid 3948 . . . . . . 7 |- ( ( B X. _V ) \ ( B X. _V ) ) = (/)
64, 5syl6sseq 3651 . . . . . 6 |- ( A C_ ( B X. _V ) -> ( A \ ( B X. _V ) ) C_ (/) )
7 ss0 3974 . . . . . 6 |- ( ( A \ ( B X. _V ) ) C_ (/) -> ( A \ ( B X. _V ) ) = (/) )
86, 7syl 17 . . . . 5 |- ( A C_ ( B X. _V ) -> ( A \ ( B X. _V ) ) = (/) )
98uneq2d 3767 . . . 4 |- ( A C_ ( B X. _V ) -> ( ( A \ C ) u. ( A \ ( B X. _V ) ) ) = ( ( A \ C ) u. (/) ) )
103, 9syl5eq 2668 . . 3 |- ( A C_ ( B X. _V ) -> ( A \ ( C i^i ( B X. _V ) ) ) = ( ( A \ C ) u. (/) ) )
11 un0 3967 . . 3 |- ( ( A \ C ) u. (/) ) = ( A \ C )
1210, 11syl6eq 2672 . 2 |- ( A C_ ( B X. _V ) -> ( A \ ( C i^i ( B X. _V ) ) ) = ( A \ C ) )
132, 12syl5eq 2668 1 |- ( A C_ ( B X. _V ) -> ( A \ ( C |` B ) ) = ( A \ C ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   = wceq 1483   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   X. cxp 5112   |` cres 5116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-res 5126
This theorem is referenced by:  qtophaus  29903
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