Theorem restidsing 5114

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 5086	. 2 |- Rel ( _I |` { A } )
2		relxp 4879	. 2 |- Rel ( { A } X. { A } )
3		velsn 3722	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 3722	. . . . 5 |- ( y e. { A } <-> y = A )
5	3, 4	anbi12i 464	. . . 4 |- ( ( x e. { A } /\ y e. { A } ) <-> ( x = A /\ y = A ) )
6		vex 2824	. . . . . . 7 |- y e. _V
7	6	ideq 4927	. . . . . 6 |- ( x _I y <-> x = y )
8	3, 7	anbi12i 464	. . . . 5 |- ( ( x e. { A } /\ x _I y ) <-> ( x = A /\ x = y ) )
9		eqeq1 2245	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
10		eqcom 2240	. . . . . . 7 |- ( A = y <-> y = A )
11	9, 10	bitrdi 196	. . . . . 6 |- ( x = A -> ( x = y <-> y = A ) )
12	11	pm5.32i 458	. . . . 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 4126	. . . . 5 |- ( x _I y <-> <. x , y >. e. _I )
15	14	anbi2i 461	. . . 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 5063	. . . 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 4799	. . 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 4864	1 |- ( _I |` { A } ) = ( { A } X. { A } )

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   {csn 3705   <.cop 3708   class class class wbr 4125   _I cid 4428   X. cxp 4767   |` cres 4771

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-res 4781

This theorem is referenced by:  grp1inv  13889