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Theorem dfsymrels5 37056
Description: Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
Assertion
Ref Expression
dfsymrels5 SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}
Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem dfsymrels5
StepHypRef Expression
1 dfsymrels4 37055 . 2 SymRels = {𝑟 ∈ Rels ∣ 𝑟 = 𝑟}
2 elrelscnveq2 37001 . 2 (𝑟 ∈ Rels → (𝑟 = 𝑟 ↔ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)))
31, 2rabimbieq 36757 1 SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1540   = wceq 1542  {crab 3406   class class class wbr 5106  ccnv 5633   Rels crels 36682   SymRels csymrels 36691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-rels 36993  df-ssr 37006  df-syms 37050  df-symrels 37051
This theorem is referenced by:  elsymrels5  37064
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