Theorem ressn 5284

Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Assertion

Ref	Expression
ressn	|- ( A |` { B } ) = ( { B } X. ( A " { B } ) )

Proof of Theorem ressn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 5047	. 2 |- Rel ( A |` { B } )
2		relxp 4841	. 2 |- Rel ( { B } X. ( A " { B } ) )
3		ancom 266	. . . 4 |- ( ( <. x , y >. e. A /\ x e. { B } ) <-> ( x e. { B } /\ <. x , y >. e. A ) )
4		vex 2806	. . . . . . 7 |- x e. _V
5		vex 2806	. . . . . . 7 |- y e. _V
6	4, 5	elimasn 5110	. . . . . 6 |- ( y e. ( A " { x } ) <-> <. x , y >. e. A )
7		elsni 3691	. . . . . . . . 9 |- ( x e. { B } -> x = B )
8	7	sneqd 3686	. . . . . . . 8 |- ( x e. { B } -> { x } = { B } )
9	8	imaeq2d 5082	. . . . . . 7 |- ( x e. { B } -> ( A " { x } ) = ( A " { B } ) )
10	9	eleq2d 2301	. . . . . 6 |- ( x e. { B } -> ( y e. ( A " { x } ) <-> y e. ( A " { B } ) ) )
11	6, 10	bitr3id 194	. . . . 5 |- ( x e. { B } -> ( <. x , y >. e. A <-> y e. ( A " { B } ) ) )
12	11	pm5.32i 454	. . . 4 |- ( ( x e. { B } /\ <. x , y >. e. A ) <-> ( x e. { B } /\ y e. ( A " { B } ) ) )
13	3, 12	bitri 184	. . 3 |- ( ( <. x , y >. e. A /\ x e. { B } ) <-> ( x e. { B } /\ y e. ( A " { B } ) ) )
14	5	opelres 5024	. . 3 |- ( <. x , y >. e. ( A |` { B } ) <-> ( <. x , y >. e. A /\ x e. { B } ) )
15		opelxp 4761	. . 3 |- ( <. x , y >. e. ( { B } X. ( A " { B } ) ) <-> ( x e. { B } /\ y e. ( A " { B } ) ) )
16	13, 14, 15	3bitr4i 212	. 2 |- ( <. x , y >. e. ( A |` { B } ) <-> <. x , y >. e. ( { B } X. ( A " { B } ) ) )
17	1, 2, 16	eqrelriiv 4826	1 |- ( A |` { B } ) = ( { B } X. ( A " { B } ) )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2202   {csn 3673   <.cop 3676   X. cxp 4729   |` cres 4733   "cima 4734

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by: (None)