Theorem cnvi 5537

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 3203	. . . . 5 |- x e. _V
2	1	ideq 5274	. . . 4 |- ( y _I x <-> y = x )
3		equcom 1945	. . . 4 |- ( y = x <-> x = y )
4	2, 3	bitri 264	. . 3 |- ( y _I x <-> x = y )
5	4	opabbii 4717	. 2 |- { <. x , y >. | y _I x } = { <. x , y >. | x = y }
6		df-cnv 5122	. 2 |- `' _I = { <. x , y >. | y _I x }
7		df-id 5024	. 2 |- _I = { <. x , y >. | x = y }
8	5, 6, 7	3eqtr4i 2654	1 |- `' _I = _I

Syntax hints:   = wceq 1483   class class class wbr 4653   {copab 4712   _I cid 5023   `'ccnv 5113

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-cnv 5122

This theorem is referenced by:  coi2  5652  funi  5920  cnvresid  5968  fcoi1  6078  ssdomg  8001  mbfid  23403  mthmpps  31479  brid  34077  extid  34081