Theorem restidsing 5012

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 4984	. 2 |- Rel ( _I |` { A } )
2		relxp 4782	. 2 |- Rel ( { A } X. { A } )
3		velsn 3649	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 3649	. . . . 5 |- ( y e. { A } <-> y = A )
5	3, 4	anbi12i 460	. . . 4 |- ( ( x e. { A } /\ y e. { A } ) <-> ( x = A /\ y = A ) )
6		vex 2774	. . . . . . 7 |- y e. _V
7	6	ideq 4828	. . . . . 6 |- ( x _I y <-> x = y )
8	3, 7	anbi12i 460	. . . . 5 |- ( ( x e. { A } /\ x _I y ) <-> ( x = A /\ x = y ) )
9		eqeq1 2211	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
10		eqcom 2206	. . . . . . 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 4044	. . . . 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 4961	. . . 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 4703	. . 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 4767	1 |- ( _I |` { A } ) = ( { A } X. { A } )

Syntax hints:   /\ wa 104   = wceq 1372   e. wcel 2175   {csn 3632   <.cop 3635   class class class wbr 4043   _I cid 4333   X. cxp 4671   |` cres 4675

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-id 4338  df-xp 4679  df-rel 4680  df-res 4685

This theorem is referenced by:  grp1inv  13357