ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvsym GIF version

Theorem cnvsym 4802
Description: Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvsym (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvsym
StepHypRef Expression
1 alcom 1412 . 2 (∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
2 relcnv 4797 . . 3 Rel 𝑅
3 ssrel 4514 . . 3 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅)))
42, 3ax-mp 7 . 2 (𝑅𝑅 ↔ ∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
5 vex 2622 . . . . . 6 𝑦 ∈ V
6 vex 2622 . . . . . 6 𝑥 ∈ V
75, 6brcnv 4607 . . . . 5 (𝑦𝑅𝑥𝑥𝑅𝑦)
8 df-br 3838 . . . . 5 (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
97, 8bitr3i 184 . . . 4 (𝑥𝑅𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
10 df-br 3838 . . . 4 (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
119, 10imbi12i 237 . . 3 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
12112albii 1405 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
131, 4, 123bitr4i 210 1 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wcel 1438  wss 2997  cop 3444   class class class wbr 3837  ccnv 4427  Rel wrel 4433
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436
This theorem is referenced by:  dfer2  6273
  Copyright terms: Public domain W3C validator