Theorem cnvi 5008

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 2729	. . . . 5 |- x e. _V
2	1	ideq 4756	. . . 4 |- ( y _I x <-> y = x )
3		equcom 1694	. . . 4 |- ( y = x <-> x = y )
4	2, 3	bitri 183	. . 3 |- ( y _I x <-> x = y )
5	4	opabbii 4049	. 2 |- { <. x , y >. | y _I x } = { <. x , y >. | x = y }
6		df-cnv 4612	. 2 |- `' _I = { <. x , y >. | y _I x }
7		df-id 4271	. 2 |- _I = { <. x , y >. | x = y }
8	5, 6, 7	3eqtr4i 2196	1 |- `' _I = _I

Syntax hints:   = wceq 1343   class class class wbr 3982   {copab 4042   _I cid 4266   `'ccnv 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612

This theorem is referenced by:  coi2  5120  funi  5220  cnvresid  5262  fcoi1  5368  ssdomg  6744