Theorem ecinxp 6666

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 3764	. . . . . . 7 |- ( ( ( R " A ) C_ A /\ B e. A ) -> { B } C_ A )
3		df-ss 3167	. . . . . . 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 5006	. . . . 5 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " ( { B } i^i A ) ) = ( R " { B } ) )
6	5	ineq1d 3360	. . . 4 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( ( R " ( { B } i^i A ) ) i^i A ) = ( ( R " { B } ) i^i A ) )
7		imass2 5042	. . . . . . 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 3190	. . . . 5 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) C_ A )
11		df-ss 3167	. . . . 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 2227	. . 3 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) = ( ( R " ( { B } i^i A ) ) i^i A ) )
14		imainrect 5112	. . 3 |- ( ( R i^i ( A X. A ) ) " { B } ) = ( ( R " ( { B } i^i A ) ) i^i A )
15	13, 14	eqtr4di 2244	. 2 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) = ( ( R i^i ( A X. A ) ) " { B } ) )
16		df-ec 6591	. 2 |- [ B ] R = ( R " { B } )
17		df-ec 6591	. 2 |- [ B ] ( R i^i ( A X. A ) ) = ( ( R i^i ( A X. A ) ) " { B } )
18	15, 16, 17	3eqtr4g 2251	1 |- ( ( ( R " A ) C_ A /\ B e. A ) -> [ B ] R = [ B ] ( R i^i ( A X. A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   i^i cin 3153   C_ wss 3154   {csn 3619   X. cxp 4658   "cima 4663   [cec 6587

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-ec 6591

This theorem is referenced by:  qsinxp  6667  nqnq0pi  7500  qusin  12912