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Theorem cnvi 5073
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
StepHypRef Expression
1 vex 2766 . . . . 5  |-  x  e. 
_V
21ideq 4824 . . . 4  |-  ( y  _I  x  <->  y  =  x )
3 equcom 1824 . . . 4  |-  ( y  =  x  <->  x  =  y )
42, 3bitri 242 . . 3  |-  ( y  _I  x  <->  x  =  y )
54opabbii 4057 . 2  |-  { <. x ,  y >.  |  y  _I  x }  =  { <. x ,  y
>.  |  x  =  y }
6 df-cnv 4677 . 2  |-  `'  _I  =  { <. x ,  y
>.  |  y  _I  x }
7 df-id 4281 . 2  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
85, 6, 73eqtr4i 2288 1  |-  `'  _I  =  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1619   class class class wbr 3997   {copab 4050    _I cid 4276   `'ccnv 4660
This theorem is referenced by:  coi2  5176  funi  5223  cnvresid  5260  fcoi1  5353  ssdomg  6875  mbfid  18953
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677
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