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Theorem cnvi 4790
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.
StepHypRef Expression
1 vex 2615 . . . . 5 𝑥 ∈ V
21ideq 4546 . . . 4 (𝑦 I 𝑥𝑦 = 𝑥)
3 equcom 1635 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
42, 3bitri 182 . . 3 (𝑦 I 𝑥𝑥 = 𝑦)
54opabbii 3871 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6 df-cnv 4409 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7 df-id 4084 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
85, 6, 73eqtr4i 2113 1 I = I
Colors of variables: wff set class
Syntax hints:   = wceq 1285   class class class wbr 3811  {copab 3864   I cid 4079  ccnv 4400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409
This theorem is referenced by:  coi2  4901  funi  4999  cnvresid  5041  fcoi1  5139  ssdomg  6425
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