Theorem restidsing 5458

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

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 5426	. 2 |- Rel ( _I |` { A } )
2		relxp 5227	. 2 |- Rel ( { A } X. { A } )
3		velsn 4193	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 4193	. . . . 5 |- ( y e. { A } <-> y = A )
5	3, 4	anbi12i 733	. . . 4 |- ( ( x e. { A } /\ y e. { A } ) <-> ( x = A /\ y = A ) )
6		vex 3203	. . . . . . 7 |- y e. _V
7	6	ideq 5274	. . . . . 6 |- ( x _I y <-> x = y )
8	7, 3	anbi12i 733	. . . . 5 |- ( ( x _I y /\ x e. { A } ) <-> ( x = y /\ x = A ) )
9		ancom 466	. . . . 5 |- ( ( x = y /\ x = A ) <-> ( x = A /\ x = y ) )
10		eqeq1 2626	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
11		eqcom 2629	. . . . . . 7 |- ( A = y <-> y = A )
12	10, 11	syl6bb 276	. . . . . 6 |- ( x = A -> ( x = y <-> y = A ) )
13	12	pm5.32i 669	. . . . 5 |- ( ( x = A /\ x = y ) <-> ( x = A /\ y = A ) )
14	8, 9, 13	3bitri 286	. . . 4 |- ( ( x _I y /\ x e. { A } ) <-> ( x = A /\ y = A ) )
15		df-br 4654	. . . . 5 |- ( x _I y <-> <. x , y >. e. _I )
16	15	anbi1i 731	. . . 4 |- ( ( x _I y /\ x e. { A } ) <-> ( <. x , y >. e. _I /\ x e. { A } ) )
17	5, 14, 16	3bitr2ri 289	. . 3 |- ( ( <. x , y >. e. _I /\ x e. { A } ) <-> ( x e. { A } /\ y e. { A } ) )
18	6	opelres 5401	. . 3 |- ( <. x , y >. e. ( _I |` { A } ) <-> ( <. x , y >. e. _I /\ x e. { A } ) )
19		opelxp 5146	. . 3 |- ( <. x , y >. e. ( { A } X. { A } ) <-> ( x e. { A } /\ y e. { A } ) )
20	17, 18, 19	3bitr4i 292	. 2 |- ( <. x , y >. e. ( _I |` { A } ) <-> <. x , y >. e. ( { A } X. { A } ) )
21	1, 2, 20	eqrelriiv 5214	1 |- ( _I |` { A } ) = ( { A } X. { A } )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   class class class wbr 4653   _I cid 5023   X. cxp 5112   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by:  residpr  6409  grp1inv  17523  psgnsn  17940  m1detdiag  20403