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

Step	Hyp	Ref	Expression
1		df-res 4862	. . 3 |- ( C |` B ) = ( C i^i ( B X. _V ) )
2	1	difeq2i 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 ) ) = (/)
6	4, 5	syl6sseq 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 ) ) = (/) )
8	6, 7	syl 17	. . . . 5 |- ( A C_ ( B X. _V ) -> ( A \ ( B X. _V ) ) = (/) )
9	8	uneq2d 3620	. . . 4 |- ( A C_ ( B X. _V ) -> ( ( A \ C ) u. ( A \ ( B X. _V ) ) ) = ( ( A \ C ) u. (/) ) )
10	3, 9	syl5eq 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 )
12	10, 11	syl6eq 2479	. 2 |- ( A C_ ( B X. _V ) -> ( A \ ( C i^i ( B X. _V ) ) ) = ( A \ C ) )
13	2, 12	syl5eq 2475	1 |- ( A C_ ( B X. _V ) -> ( A \ ( C |` B ) ) = ( A \ C ) )

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