Theorem cnvi 5033

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 2740	. . . . 5 |- x e. _V
2	1	ideq 4779	. . . 4 |- ( y _I x <-> y = x )
3		equcom 1706	. . . 4 |- ( y = x <-> x = y )
4	2, 3	bitri 184	. . 3 |- ( y _I x <-> x = y )
5	4	opabbii 4070	. 2 |- { <. x , y >. | y _I x } = { <. x , y >. | x = y }
6		df-cnv 4634	. 2 |- `' _I = { <. x , y >. | y _I x }
7		df-id 4293	. 2 |- _I = { <. x , y >. | x = y }
8	5, 6, 7	3eqtr4i 2208	1 |- `' _I = _I

Syntax hints:   = wceq 1353   class class class wbr 4003   {copab 4063   _I cid 4288   `'ccnv 4625

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634

This theorem is referenced by:  coi2  5145  funi  5248  cnvresid  5290  fcoi1  5396  ssdomg  6777