Theorem cnvi 4951

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 2692	. . . . 5 |- x e. _V
2	1	ideq 4699	. . . 4 |- ( y _I x <-> y = x )
3		equcom 1683	. . . 4 |- ( y = x <-> x = y )
4	2, 3	bitri 183	. . 3 |- ( y _I x <-> x = y )
5	4	opabbii 4003	. 2 |- { <. x , y >. | y _I x } = { <. x , y >. | x = y }
6		df-cnv 4555	. 2 |- `' _I = { <. x , y >. | y _I x }
7		df-id 4223	. 2 |- _I = { <. x , y >. | x = y }
8	5, 6, 7	3eqtr4i 2171	1 |- `' _I = _I

Syntax hints:   = wceq 1332   class class class wbr 3937   {copab 3996   _I cid 4218   `'ccnv 4546

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555

This theorem is referenced by:  coi2  5063  funi  5163  cnvresid  5205  fcoi1  5311  ssdomg  6680