Theorem ecinxp 6710

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 3784	. . . . . . 7 |- ( ( ( R " A ) C_ A /\ B e. A ) -> { B } C_ A )
3		df-ss 3183	. . . . . . 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 5031	. . . . 5 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " ( { B } i^i A ) ) = ( R " { B } ) )
6	5	ineq1d 3377	. . . 4 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( ( R " ( { B } i^i A ) ) i^i A ) = ( ( R " { B } ) i^i A ) )
7		imass2 5067	. . . . . . 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 3207	. . . . 5 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) C_ A )
11		df-ss 3183	. . . . 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 2240	. . 3 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) = ( ( R " ( { B } i^i A ) ) i^i A ) )
14		imainrect 5137	. . 3 |- ( ( R i^i ( A X. A ) ) " { B } ) = ( ( R " ( { B } i^i A ) ) i^i A )
15	13, 14	eqtr4di 2257	. 2 |- ( ( ( R " A ) C_ A /\ B e. A ) -> ( R " { B } ) = ( ( R i^i ( A X. A ) ) " { B } ) )
16		df-ec 6635	. 2 |- [ B ] R = ( R " { B } )
17		df-ec 6635	. 2 |- [ B ] ( R i^i ( A X. A ) ) = ( ( R i^i ( A X. A ) ) " { B } )
18	15, 16, 17	3eqtr4g 2264	1 |- ( ( ( R " A ) C_ A /\ B e. A ) -> [ B ] R = [ B ] ( R i^i ( A X. A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   i^i cin 3169   C_ wss 3170   {csn 3638   X. cxp 4681   "cima 4686   [cec 6631

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-xp 4689  df-rel 4690  df-cnv 4691  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-ec 6635

This theorem is referenced by:  qsinxp  6711  nqnq0pi  7571  qusin  13233