Theorem cnvi 4748

Description: The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
cnvi	|- `' _I = _I

Proof of Theorem cnvi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . . 5 |- x e. _V
2	1	ideq 4506	. . . 4 |- ( y _I x <-> y = x )
3		equcom 1633	. . . 4 |- ( y = x <-> x = y )
4	2, 3	bitri 182	. . 3 |- ( y _I x <-> x = y )
5	4	opabbii 3845	. 2 |- { <. x , y >. | y _I x } = { <. x , y >. | x = y }
6		df-cnv 4371	. 2 |- `' _I = { <. x , y >. | y _I x }
7		df-id 4048	. 2 |- _I = { <. x , y >. | x = y }
8	5, 6, 7	3eqtr4i 2111	1 |- `' _I = _I

Syntax hints:   = wceq 1284   class class class wbr 3785   {copab 3838   _I cid 4043   `'ccnv 4362

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371

This theorem is referenced by:  coi2  4857  funi  4952  cnvresid  4993  fcoi1  5090  ssdomg  6281