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Theorem dfsymrels3 38527
Description: Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
Assertion
Ref Expression
dfsymrels3 SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}
Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem dfsymrels3
StepHypRef Expression
1 dfsymrels2 38526 . 2 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
2 cnvsym 6134 . 2 (𝑟𝑟 ↔ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))
31, 2rabbieq 3441 1 SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1534   = wceq 1536  {crab 3432  wss 3962   class class class wbr 5147  ccnv 5687   Rels crels 38163   SymRels csymrels 38172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-rels 38466  df-ssr 38479  df-syms 38523  df-symrels 38524
This theorem is referenced by:  elsymrels3  38535
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