Theorem dftrrels3 38578

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

Assertion

Ref	Expression
dftrrels3	⊢ TrRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)}

Distinct variable group:   𝑥,𝑟,𝑦,𝑧

Proof of Theorem dftrrels3

Step	Hyp	Ref	Expression
1		dftrrels2 38577	. 2 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
2		cotr 6129	. 2 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))
3	1, 2	rabbieq 3444	1 ⊢ TrRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)}

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539  {crab 3435   ⊆ wss 3950   class class class wbr 5142   ∘ ccom 5688   Rels crels 38185   TrRels ctrrels 38197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-rels 38487  df-ssr 38500  df-trs 38574  df-trrels 38575

This theorem is referenced by:  eltrrels3  38582