Theorem cnvi 4728

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 2560	. . . . 5 |- x e. _V
2	1	ideq 4488	. . . 4 |- ( y _I x <-> y = x )
3		equcom 1593	. . . 4 |- ( y = x <-> x = y )
4	2, 3	bitri 173	. . 3 |- ( y _I x <-> x = y )
5	4	opabbii 3824	. 2 |- { <. x , y >. | y _I x } = { <. x , y >. | x = y }
6		df-cnv 4353	. 2 |- `' _I = { <. x , y >. | y _I x }
7		df-id 4030	. 2 |- _I = { <. x , y >. | x = y }
8	5, 6, 7	3eqtr4i 2070	1 |- `' _I = _I

Syntax hints:   = wceq 1243   class class class wbr 3764   {copab 3817   _I cid 4025   `'ccnv 4344

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353

This theorem is referenced by:  coi2  4837  funi  4932  cnvresid  4973  fcoi1  5070  ssdomg  6258