Theorem restidsing 4948

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 4920	. 2 |- Rel ( _I |` { A } )
2		relxp 4721	. 2 |- Rel ( { A } X. { A } )
3		velsn 3601	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 3601	. . . . 5 |- ( y e. { A } <-> y = A )
5	3, 4	anbi12i 458	. . . 4 |- ( ( x e. { A } /\ y e. { A } ) <-> ( x = A /\ y = A ) )
6		vex 2734	. . . . . . 7 |- y e. _V
7	6	ideq 4764	. . . . . 6 |- ( x _I y <-> x = y )
8	3, 7	anbi12i 458	. . . . 5 |- ( ( x e. { A } /\ x _I y ) <-> ( x = A /\ x = y ) )
9		eqeq1 2178	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
10		eqcom 2173	. . . . . . 7 |- ( A = y <-> y = A )
11	9, 10	bitrdi 195	. . . . . 6 |- ( x = A -> ( x = y <-> y = A ) )
12	11	pm5.32i 452	. . . . 5 |- ( ( x = A /\ x = y ) <-> ( x = A /\ y = A ) )
13	8, 12	bitri 183	. . . 4 |- ( ( x e. { A } /\ x _I y ) <-> ( x = A /\ y = A ) )
14		df-br 3991	. . . . 5 |- ( x _I y <-> <. x , y >. e. _I )
15	14	anbi2i 455	. . . 4 |- ( ( x e. { A } /\ x _I y ) <-> ( x e. { A } /\ <. x , y >. e. _I ) )
16	5, 13, 15	3bitr2ri 208	. . 3 |- ( ( x e. { A } /\ <. x , y >. e. _I ) <-> ( x e. { A } /\ y e. { A } ) )
17	6	opelres 4897	. . . 4 |- ( <. x , y >. e. ( _I |` { A } ) <-> ( <. x , y >. e. _I /\ x e. { A } ) )
18	17	biancomi 268	. . 3 |- ( <. x , y >. e. ( _I |` { A } ) <-> ( x e. { A } /\ <. x , y >. e. _I ) )
19		opelxp 4642	. . 3 |- ( <. x , y >. e. ( { A } X. { A } ) <-> ( x e. { A } /\ y e. { A } ) )
20	16, 18, 19	3bitr4i 211	. 2 |- ( <. x , y >. e. ( _I |` { A } ) <-> <. x , y >. e. ( { A } X. { A } ) )
21	1, 2, 20	eqrelriiv 4706	1 |- ( _I |` { A } ) = ( { A } X. { A } )

Syntax hints:   /\ wa 103   = wceq 1349   e. wcel 2142   {csn 3584   <.cop 3587   class class class wbr 3990   _I cid 4274   X. cxp 4610   |` cres 4614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-br 3991  df-opab 4052  df-id 4279  df-xp 4618  df-rel 4619  df-res 4624

This theorem is referenced by:  grp1inv  12828