Theorem cnvi 4943

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 2689	. . . . 5 |- x e. _V
2	1	ideq 4691	. . . 4 |- ( y _I x <-> y = x )
3		equcom 1682	. . . 4 |- ( y = x <-> x = y )
4	2, 3	bitri 183	. . 3 |- ( y _I x <-> x = y )
5	4	opabbii 3995	. 2 |- { <. x , y >. | y _I x } = { <. x , y >. | x = y }
6		df-cnv 4547	. 2 |- `' _I = { <. x , y >. | y _I x }
7		df-id 4215	. 2 |- _I = { <. x , y >. | x = y }
8	5, 6, 7	3eqtr4i 2170	1 |- `' _I = _I

Syntax hints:   = wceq 1331   class class class wbr 3929   {copab 3988   _I cid 4210   `'ccnv 4538

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547

This theorem is referenced by:  coi2  5055  funi  5155  cnvresid  5197  fcoi1  5303  ssdomg  6672