Theorem dftrrels3 38981

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 38980	. 2 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
2		cotr 6075	. 2 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))
3	1, 2	rabbieq 3397	1 ⊢ TrRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)}

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  {crab 3389   ⊆ wss 3889   class class class wbr 5085   ∘ ccom 5635   Rels crels 38506   TrRels ctrrels 38518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-rels 38761  df-ssr 38899  df-trs 38977  df-trrels 38978

This theorem is referenced by:  eltrrels3  38985