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Theorem elrelscnveq3 38015
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
elrelscnveq3 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrelscnveq3
StepHypRef Expression
1 eqss 3989 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
2 cnvsym 6114 . . . . . . 7 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
32biimpi 215 . . . . . 6 (𝑅𝑅 → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
43a1d 25 . . . . 5 (𝑅𝑅 → (𝑅 ∈ Rels → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
54adantl 480 . . . 4 ((𝑅𝑅𝑅𝑅) → (𝑅 ∈ Rels → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
65com12 32 . . 3 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
7 elrelsrelim 38012 . . . . . 6 (𝑅 ∈ Rels → Rel 𝑅)
8 dfrel2 6189 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
97, 8sylib 217 . . . . 5 (𝑅 ∈ Rels → 𝑅 = 𝑅)
10 cnvss 5870 . . . . . . 7 (𝑅𝑅𝑅𝑅)
11 sseq1 3999 . . . . . . 7 (𝑅 = 𝑅 → (𝑅𝑅𝑅𝑅))
1210, 11syl5ibcom 244 . . . . . 6 (𝑅𝑅 → (𝑅 = 𝑅𝑅𝑅))
132, 12sylbir 234 . . . . 5 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (𝑅 = 𝑅𝑅𝑅))
149, 13syl5com 31 . . . 4 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅))
152biimpri 227 . . . 4 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅)
1614, 15jca2 512 . . 3 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (𝑅𝑅𝑅𝑅)))
176, 16impbid 211 . 2 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
181, 17bitrid 282 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 3941   class class class wbr 5144  ccnv 5672  Rel wrel 5678   Rels crels 37703
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 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
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 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-xp 5679  df-rel 5680  df-cnv 5681  df-rels 38009
This theorem is referenced by:  elrelscnveq  38016
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