Theorem restidsingOLD 5459

Description: Obsolete proof of restidsing 5458 as of 25-Aug-2021. (Contributed by FL, 2-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
restidsingOLD	|- ( _I |` { A } ) = ( { A } X. { A } )

Proof of Theorem restidsingOLD

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		df-br 4654	. . . . . 6 |- ( x _I y <-> <. x , y >. e. _I )
4	3	bicomi 214	. . . . 5 |- ( <. x , y >. e. _I <-> x _I y )
5	4	anbi1i 731	. . . 4 |- ( ( <. x , y >. e. _I /\ x e. { A } ) <-> ( x _I y /\ x e. { A } ) )
6		simpr 477	. . . . . 6 |- ( ( x _I y /\ x e. { A } ) -> x e. { A } )
7		velsn 4193	. . . . . . . 8 |- ( x e. { A } <-> x = A )
8		vex 3203	. . . . . . . . . 10 |- y e. _V
9		ideqg 5273	. . . . . . . . . . 11 |- ( y e. _V -> ( x _I y <-> x = y ) )
10	9	biimpd 219	. . . . . . . . . 10 |- ( y e. _V -> ( x _I y -> x = y ) )
11	8, 10	ax-mp 5	. . . . . . . . 9 |- ( x _I y -> x = y )
12		eqtr2 2642	. . . . . . . . . . 11 |- ( ( x = y /\ x = A ) -> y = A )
13	12	ex 450	. . . . . . . . . 10 |- ( x = y -> ( x = A -> y = A ) )
14		velsn 4193	. . . . . . . . . 10 |- ( y e. { A } <-> y = A )
15	13, 14	syl6ibr 242	. . . . . . . . 9 |- ( x = y -> ( x = A -> y e. { A } ) )
16	11, 15	syl 17	. . . . . . . 8 |- ( x _I y -> ( x = A -> y e. { A } ) )
17	7, 16	syl5bi 232	. . . . . . 7 |- ( x _I y -> ( x e. { A } -> y e. { A } ) )
18	17	imp 445	. . . . . 6 |- ( ( x _I y /\ x e. { A } ) -> y e. { A } )
19	6, 18	jca 554	. . . . 5 |- ( ( x _I y /\ x e. { A } ) -> ( x e. { A } /\ y e. { A } ) )
20		eqtr3 2643	. . . . . . . . . . . 12 |- ( ( y = A /\ x = A ) -> y = x )
21	8	ideq 5274	. . . . . . . . . . . . 13 |- ( x _I y <-> x = y )
22		equcom 1945	. . . . . . . . . . . . 13 |- ( x = y <-> y = x )
23	21, 22	bitri 264	. . . . . . . . . . . 12 |- ( x _I y <-> y = x )
24	20, 23	sylibr 224	. . . . . . . . . . 11 |- ( ( y = A /\ x = A ) -> x _I y )
25	24	ex 450	. . . . . . . . . 10 |- ( y = A -> ( x = A -> x _I y ) )
26	14, 25	sylbi 207	. . . . . . . . 9 |- ( y e. { A } -> ( x = A -> x _I y ) )
27	26	com12 32	. . . . . . . 8 |- ( x = A -> ( y e. { A } -> x _I y ) )
28	7, 27	sylbi 207	. . . . . . 7 |- ( x e. { A } -> ( y e. { A } -> x _I y ) )
29	28	imp 445	. . . . . 6 |- ( ( x e. { A } /\ y e. { A } ) -> x _I y )
30		simpl 473	. . . . . 6 |- ( ( x e. { A } /\ y e. { A } ) -> x e. { A } )
31	29, 30	jca 554	. . . . 5 |- ( ( x e. { A } /\ y e. { A } ) -> ( x _I y /\ x e. { A } ) )
32	19, 31	impbii 199	. . . 4 |- ( ( x _I y /\ x e. { A } ) <-> ( x e. { A } /\ y e. { A } ) )
33	5, 32	bitri 264	. . 3 |- ( ( <. x , y >. e. _I /\ x e. { A } ) <-> ( x e. { A } /\ y e. { A } ) )
34	8	opelres 5401	. . 3 |- ( <. x , y >. e. ( _I |` { A } ) <-> ( <. x , y >. e. _I /\ x e. { A } ) )
35		opelxp 5146	. . 3 |- ( <. x , y >. e. ( { A } X. { A } ) <-> ( x e. { A } /\ y e. { A } ) )
36	33, 34, 35	3bitr4i 292	. 2 |- ( <. x , y >. e. ( _I |` { A } ) <-> <. x , y >. e. ( { A } X. { A } ) )
37	1, 2, 36	eqrelriiv 5214	1 |- ( _I |` { A } ) = ( { A } X. { A } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {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: (None)