Theorem ecinxp 6874

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 3855	. . . . . . 7 |- ( ( ( R " A ) C_ A /\ B e. A ) -> { B } C_ A )
3		df-ss 3233	. . . . . . 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 5121	. . . . 5 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " ( { B } i^i A ) ) = ( R " { B } ) )
6	5	ineq1d 3431	. . . 4 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( ( R " ( { B } i^i A ) ) i^i A ) = ( ( R " { B } ) i^i A ) )
7		imass2 5158	. . . . . . 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 3258	. . . . 5 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) C_ A )
11		df-ss 3233	. . . . 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 2272	. . 3 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) = ( ( R " ( { B } i^i A ) ) i^i A ) )
14		imainrect 5228	. . 3 |- ( ( R i^i ( A X. A ) ) " { B } ) = ( ( R " ( { B } i^i A ) ) i^i A )
15	13, 14	eqtr4di 2289	. 2 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) = ( ( R i^i ( A X. A ) ) " { B } ) )
16		df-ec 6799	. 2 |- [ B ] R = ( R " { B } )
17		df-ec 6799	. 2 |- [ B ] ( R i^i ( A X. A ) ) = ( ( R i^i ( A X. A ) ) " { B } )
18	15, 16, 17	3eqtr4g 2296	1 |- ( ( ( R " A ) C_ A /\ B e. A ) -> [ B ] R = [ B ] ( R i^i ( A X. A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   i^i cin 3219   C_ wss 3220   {csn 3705   X. cxp 4767   "cima 4772   [cec 6795

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-ec 6799

This theorem is referenced by:  qsinxp  6875  nqnq0pi  7795  qusin  13624