Theorem dfsymrels3 35784

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 35783	. 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
2		cnvsym 5976	. 2 ⊢ (◡𝑟 ⊆ 𝑟 ↔ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))
3	1, 2	rabbieq 35514	1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)}

Syntax hints:   → wi 4  ∀wal 1535   = wceq 1537  {crab 3144   ⊆ wss 3938   class class class wbr 5068  ^◡ccnv 5556   Rels crels 35457   SymRels csymrels 35466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-rels 35727  df-ssr 35740  df-syms 35780  df-symrels 35781

This theorem is referenced by:  elsymrels3  35792