Theorem dfsymrels3 39126

Description: Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)

Assertion

Ref	Expression
dfsymrels3	⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)}

Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem dfsymrels3

Step	Hyp	Ref	Expression
1		dfsymrels2 39125	. 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
2		cnvsym 6102	. 2 ⊢ (◡𝑟 ⊆ 𝑟 ↔ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))
3	1, 2	rabbieq 3423	1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)}

Syntax hints:   → wi 4  ∀wal 1559   = wceq 1561  {crab 3415   ⊆ wss 3905   class class class wbr 5101  ^◡ccnv 5647   Rels crels 38685   SymRels csymrels 38694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-rel 5655  df-cnv 5656  df-dm 5658  df-rn 5659  df-res 5660  df-rels 38940  df-ssr 39078  df-syms 39122  df-symrels 39123

This theorem is referenced by:  elsymrels3  39138