Theorem restidsing 5002

Description: Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) (Proof shortened by Peter Mazsa, 6-Oct-2022.)

Assertion

Ref	Expression
restidsing	|- ( _I |` { A } ) = ( { A } X. { A } )

Proof of Theorem restidsing

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

Step	Hyp	Ref	Expression
1		relres 4974	. 2 |- Rel ( _I |` { A } )
2		relxp 4772	. 2 |- Rel ( { A } X. { A } )
3		velsn 3639	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 3639	. . . . 5 |- ( y e. { A } <-> y = A )
5	3, 4	anbi12i 460	. . . 4 |- ( ( x e. { A } /\ y e. { A } ) <-> ( x = A /\ y = A ) )
6		vex 2766	. . . . . . 7 |- y e. _V
7	6	ideq 4818	. . . . . 6 |- ( x _I y <-> x = y )
8	3, 7	anbi12i 460	. . . . 5 |- ( ( x e. { A } /\ x _I y ) <-> ( x = A /\ x = y ) )
9		eqeq1 2203	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
10		eqcom 2198	. . . . . . 7 |- ( A = y <-> y = A )
11	9, 10	bitrdi 196	. . . . . 6 |- ( x = A -> ( x = y <-> y = A ) )
12	11	pm5.32i 454	. . . . 5 |- ( ( x = A /\ x = y ) <-> ( x = A /\ y = A ) )
13	8, 12	bitri 184	. . . 4 |- ( ( x e. { A } /\ x _I y ) <-> ( x = A /\ y = A ) )
14		df-br 4034	. . . . 5 |- ( x _I y <-> <. x , y >. e. _I )
15	14	anbi2i 457	. . . 4 |- ( ( x e. { A } /\ x _I y ) <-> ( x e. { A } /\ <. x , y >. e. _I ) )
16	5, 13, 15	3bitr2ri 209	. . 3 |- ( ( x e. { A } /\ <. x , y >. e. _I ) <-> ( x e. { A } /\ y e. { A } ) )
17	6	opelres 4951	. . . 4 |- ( <. x , y >. e. ( _I |` { A } ) <-> ( <. x , y >. e. _I /\ x e. { A } ) )
18	17	biancomi 270	. . 3 |- ( <. x , y >. e. ( _I |` { A } ) <-> ( x e. { A } /\ <. x , y >. e. _I ) )
19		opelxp 4693	. . 3 |- ( <. x , y >. e. ( { A } X. { A } ) <-> ( x e. { A } /\ y e. { A } ) )
20	16, 18, 19	3bitr4i 212	. 2 |- ( <. x , y >. e. ( _I |` { A } ) <-> <. x , y >. e. ( { A } X. { A } ) )
21	1, 2, 20	eqrelriiv 4757	1 |- ( _I |` { A } ) = ( { A } X. { A } )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2167   {csn 3622   <.cop 3625   class class class wbr 4033   _I cid 4323   X. cxp 4661   |` cres 4665

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-res 4675

This theorem is referenced by:  grp1inv  13239