Theorem cnvi 5015

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 2733	. . . . 5 |- x e. _V
2	1	ideq 4763	. . . 4 |- ( y _I x <-> y = x )
3		equcom 1699	. . . 4 |- ( y = x <-> x = y )
4	2, 3	bitri 183	. . 3 |- ( y _I x <-> x = y )
5	4	opabbii 4056	. 2 |- { <. x , y >. | y _I x } = { <. x , y >. | x = y }
6		df-cnv 4619	. 2 |- `' _I = { <. x , y >. | y _I x }
7		df-id 4278	. 2 |- _I = { <. x , y >. | x = y }
8	5, 6, 7	3eqtr4i 2201	1 |- `' _I = _I

Syntax hints:   = wceq 1348   class class class wbr 3989   {copab 4049   _I cid 4273   `'ccnv 4610

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619

This theorem is referenced by:  coi2  5127  funi  5230  cnvresid  5272  fcoi1  5378  ssdomg  6756