Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elrelscnveq2 Structured version   Visualization version   GIF version

Theorem elrelscnveq2 37997
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
elrelscnveq2 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrelscnveq2
StepHypRef Expression
1 cnvsym 6123 . . . 4 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
21a1i 11 . . 3 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
3 elrelsrelim 37992 . . . . . . 7 (𝑅 ∈ Rels → Rel 𝑅)
4 dfrel2 6198 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
53, 4sylib 217 . . . . . 6 (𝑅 ∈ Rels → 𝑅 = 𝑅)
65sseq1d 4013 . . . . 5 (𝑅 ∈ Rels → (𝑅𝑅𝑅𝑅))
7 cnvsym 6123 . . . . 5 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
86, 7bitr3di 285 . . . 4 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
9 relbrcnvg 6114 . . . . . . 7 (Rel 𝑅 → (𝑥𝑅𝑦𝑦𝑅𝑥))
103, 9syl 17 . . . . . 6 (𝑅 ∈ Rels → (𝑥𝑅𝑦𝑦𝑅𝑥))
11 relbrcnvg 6114 . . . . . . 7 (Rel 𝑅 → (𝑦𝑅𝑥𝑥𝑅𝑦))
123, 11syl 17 . . . . . 6 (𝑅 ∈ Rels → (𝑦𝑅𝑥𝑥𝑅𝑦))
1310, 12imbi12d 343 . . . . 5 (𝑅 ∈ Rels → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥𝑥𝑅𝑦)))
14132albidv 1918 . . . 4 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
158, 14bitrd 278 . . 3 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
162, 15anbi12d 630 . 2 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦))))
17 eqss 3997 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
18 2albiim 1885 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
1916, 17, 183bitr4g 313 1 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wcel 2098  wss 3949   class class class wbr 5152  ccnv 5681  Rel wrel 5687   Rels crels 37683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-rels 37989
This theorem is referenced by:  elrelscnveq4  37998  dfsymrels5  38052
  Copyright terms: Public domain W3C validator