Theorem restidsing 5075

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 5047	. 2 |- Rel ( _I |` { A } )
2		relxp 4841	. 2 |- Rel ( { A } X. { A } )
3		velsn 3690	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 3690	. . . . 5 |- ( y e. { A } <-> y = A )
5	3, 4	anbi12i 460	. . . 4 |- ( ( x e. { A } /\ y e. { A } ) <-> ( x = A /\ y = A ) )
6		vex 2806	. . . . . . 7 |- y e. _V
7	6	ideq 4888	. . . . . 6 |- ( x _I y <-> x = y )
8	3, 7	anbi12i 460	. . . . 5 |- ( ( x e. { A } /\ x _I y ) <-> ( x = A /\ x = y ) )
9		eqeq1 2238	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
10		eqcom 2233	. . . . . . 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 4094	. . . . 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 5024	. . . 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 4761	. . 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 4826	1 |- ( _I |` { A } ) = ( { A } X. { A } )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2202   {csn 3673   <.cop 3676   class class class wbr 4093   _I cid 4391   X. cxp 4729   |` cres 4733

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-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-id 4396  df-xp 4737  df-rel 4738  df-res 4743

This theorem is referenced by:  grp1inv  13753