Theorem ecinxp 6609

Description: Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)

Assertion

Ref	Expression
ecinxp	|- ( ( ( R " A ) C_ A /\ B e. A ) -> [ B ] R = [ B ] ( R i^i ( A X. A ) ) )

Proof of Theorem ecinxp

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . . . 8 |- ( ( ( R " A ) C_ A /\ B e. A ) -> B e. A )
2	1	snssd 3737	. . . . . . 7 |- ( ( ( R " A ) C_ A /\ B e. A ) -> { B } C_ A )
3		df-ss 3142	. . . . . . 7 |- ( { B } C_ A <-> ( { B } i^i A ) = { B } )
4	2, 3	sylib 122	. . . . . 6 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( { B } i^i A ) = { B } )
5	4	imaeq2d 4970	. . . . 5 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " ( { B } i^i A ) ) = ( R " { B } ) )
6	5	ineq1d 3335	. . . 4 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( ( R " ( { B } i^i A ) ) i^i A ) = ( ( R " { B } ) i^i A ) )
7		imass2 5004	. . . . . . 7 |- ( { B } C_ A -> ( R " { B } ) C_ ( R " A ) )
8	2, 7	syl 14	. . . . . 6 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) C_ ( R " A ) )
9		simpl 109	. . . . . 6 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " A ) C_ A )
10	8, 9	sstrd 3165	. . . . 5 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) C_ A )
11		df-ss 3142	. . . . 5 |- ( ( R " { B } ) C_ A <-> ( ( R " { B } ) i^i A ) = ( R " { B } ) )
12	10, 11	sylib 122	. . . 4 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( ( R " { B } ) i^i A ) = ( R " { B } ) )
13	6, 12	eqtr2d 2211	. . 3 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) = ( ( R " ( { B } i^i A ) ) i^i A ) )
14		imainrect 5074	. . 3 |- ( ( R i^i ( A X. A ) ) " { B } ) = ( ( R " ( { B } i^i A ) ) i^i A )
15	13, 14	eqtr4di 2228	. 2 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) = ( ( R i^i ( A X. A ) ) " { B } ) )
16		df-ec 6536	. 2 |- [ B ] R = ( R " { B } )
17		df-ec 6536	. 2 |- [ B ] ( R i^i ( A X. A ) ) = ( ( R i^i ( A X. A ) ) " { B } )
18	15, 16, 17	3eqtr4g 2235	1 |- ( ( ( R " A ) C_ A /\ B e. A ) -> [ B ] R = [ B ] ( R i^i ( A X. A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   i^i cin 3128   C_ wss 3129   {csn 3592   X. cxp 4624   "cima 4629   [cec 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-ec 6536

This theorem is referenced by:  qsinxp  6610  nqnq0pi  7436