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Theorem cnvsym 5498
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 2035 . 2 (∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
2 relcnv 5491 . . 3 Rel 𝑅
3 ssrel 5197 . . 3 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅)))
42, 3ax-mp 5 . 2 (𝑅𝑅 ↔ ∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
5 vex 3198 . . . . . 6 𝑦 ∈ V
6 vex 3198 . . . . . 6 𝑥 ∈ V
75, 6brcnv 5294 . . . . 5 (𝑦𝑅𝑥𝑥𝑅𝑦)
8 df-br 4645 . . . . 5 (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
97, 8bitr3i 266 . . . 4 (𝑥𝑅𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
10 df-br 4645 . . . 4 (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
119, 10imbi12i 340 . . 3 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
12112albii 1746 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
131, 4, 123bitr4i 292 1 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479  wcel 1988  wss 3567  cop 4174   class class class wbr 4644  ccnv 5103  Rel wrel 5109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112
This theorem is referenced by:  dfer2  7728
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