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Theorem cnvi 5782
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 3417 . . . . 5 𝑥 ∈ V
21ideq 5511 . . . 4 (𝑦 I 𝑥𝑦 = 𝑥)
3 equcom 2122 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
42, 3bitri 267 . . 3 (𝑦 I 𝑥𝑥 = 𝑦)
54opabbii 4942 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6 df-cnv 5354 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7 df-id 5252 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
85, 6, 73eqtr4i 2859 1 I = I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656   class class class wbr 4875  {copab 4937   I cid 5251  ccnv 5345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354
This theorem is referenced by:  coi2  5897  funi  6159  cnvresid  6205  fcoi1  6319  ssdomg  8274  mbfid  23808  mthmpps  32021  brid  34621  extid  34625  cosscnvid  34774  idsymrel  34850
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