Theorem cnvi 5141

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 2805	. . . . 5 |- x e. _V
2	1	ideq 4882	. . . 4 |- ( y _I x <-> y = x )
3		equcom 1754	. . . 4 |- ( y = x <-> x = y )
4	2, 3	bitri 184	. . 3 |- ( y _I x <-> x = y )
5	4	opabbii 4156	. 2 |- { <. x , y >. | y _I x } = { <. x , y >. | x = y }
6		df-cnv 4733	. 2 |- `' _I = { <. x , y >. | y _I x }
7		df-id 4390	. 2 |- _I = { <. x , y >. | x = y }
8	5, 6, 7	3eqtr4i 2262	1 |- `' _I = _I

Syntax hints:   = wceq 1397   class class class wbr 4088   {copab 4149   _I cid 4385   `'ccnv 4724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733

This theorem is referenced by:  coi2  5253  funi  5358  cnvresid  5404  fcoi1  5517  ssdomg  6951