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

Theorem cnvsym 4631
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 1343 . 2 (yx(⟨y, x 𝑅 → ⟨y, x 𝑅) ↔ xy(⟨y, x 𝑅 → ⟨y, x 𝑅))
2 relcnv 4626 . . 3 Rel 𝑅
3 ssrel 4351 . . 3 (Rel 𝑅 → (𝑅𝑅yx(⟨y, x 𝑅 → ⟨y, x 𝑅)))
42, 3ax-mp 7 . 2 (𝑅𝑅yx(⟨y, x 𝑅 → ⟨y, x 𝑅))
5 vex 2534 . . . . . 6 y V
6 vex 2534 . . . . . 6 x V
75, 6brcnv 4441 . . . . 5 (y𝑅xx𝑅y)
8 df-br 3735 . . . . 5 (y𝑅x ↔ ⟨y, x 𝑅)
97, 8bitr3i 175 . . . 4 (x𝑅y ↔ ⟨y, x 𝑅)
10 df-br 3735 . . . 4 (y𝑅x ↔ ⟨y, x 𝑅)
119, 10imbi12i 228 . . 3 ((x𝑅yy𝑅x) ↔ (⟨y, x 𝑅 → ⟨y, x 𝑅))
12112albii 1336 . 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 1224   wcel 1370  wss 2890  cop 3349   class class class wbr 3734  ccnv 4267  Rel wrel 4273
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276
This theorem is referenced by:  dfer2  6014
  Copyright terms: Public domain W3C validator