Theorem ecinxp 6844

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 3839	. . . . . . 7 |- ( ( ( R " A ) C_ A /\ B e. A ) -> { B } C_ A )
3		df-ss 3224	. . . . . . 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 5101	. . . . 5 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " ( { B } i^i A ) ) = ( R " { B } ) )
6	5	ineq1d 3421	. . . 4 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( ( R " ( { B } i^i A ) ) i^i A ) = ( ( R " { B } ) i^i A ) )
7		imass2 5138	. . . . . . 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 3248	. . . . 5 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) C_ A )
11		df-ss 3224	. . . . 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 2266	. . 3 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) = ( ( R " ( { B } i^i A ) ) i^i A ) )
14		imainrect 5208	. . 3 |- ( ( R i^i ( A X. A ) ) " { B } ) = ( ( R " ( { B } i^i A ) ) i^i A )
15	13, 14	eqtr4di 2283	. 2 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) = ( ( R i^i ( A X. A ) ) " { B } ) )
16		df-ec 6769	. 2 |- [ B ] R = ( R " { B } )
17		df-ec 6769	. 2 |- [ B ] ( R i^i ( A X. A ) ) = ( ( R i^i ( A X. A ) ) " { B } )
18	15, 16, 17	3eqtr4g 2290	1 |- ( ( ( R " A ) C_ A /\ B e. A ) -> [ B ] R = [ B ] ( R i^i ( A X. A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   i^i cin 3210   C_ wss 3211   {csn 3689   X. cxp 4747   "cima 4752   [cec 6765

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-ec 6769

This theorem is referenced by:  qsinxp  6845  nqnq0pi  7753  qusin  13539