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Theorem cnvsym 4651
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 (𝑅𝑅xy(x𝑅yy𝑅x))
Distinct variable group:   x,y,𝑅

Proof of Theorem cnvsym
StepHypRef Expression
1 alcom 1364 . 2 (yx(⟨y, x 𝑅 → ⟨y, x 𝑅) ↔ xy(⟨y, x 𝑅 → ⟨y, x 𝑅))
2 relcnv 4646 . . 3 Rel 𝑅
3 ssrel 4371 . . 3 (Rel 𝑅 → (𝑅𝑅yx(⟨y, x 𝑅 → ⟨y, x 𝑅)))
42, 3ax-mp 7 . 2 (𝑅𝑅yx(⟨y, x 𝑅 → ⟨y, x 𝑅))
5 vex 2554 . . . . . 6 y V
6 vex 2554 . . . . . 6 x V
75, 6brcnv 4461 . . . . 5 (y𝑅xx𝑅y)
8 df-br 3756 . . . . 5 (y𝑅x ↔ ⟨y, x 𝑅)
97, 8bitr3i 175 . . . 4 (x𝑅y ↔ ⟨y, x 𝑅)
10 df-br 3756 . . . 4 (y𝑅x ↔ ⟨y, x 𝑅)
119, 10imbi12i 228 . . 3 ((x𝑅yy𝑅x) ↔ (⟨y, x 𝑅 → ⟨y, x 𝑅))
12112albii 1357 . 2 (xy(x𝑅yy𝑅x) ↔ xy(⟨y, x 𝑅 → ⟨y, x 𝑅))
131, 4, 123bitr4i 201 1 (𝑅𝑅xy(x𝑅yy𝑅x))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   wcel 1390  wss 2911  cop 3370   class class class wbr 3755  ccnv 4287  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296
This theorem is referenced by:  dfer2  6043
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