Theorem dftrrels3 38562

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 38561	. 2 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
2		cotr 6085	. 2 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))
3	1, 2	rabbieq 3417	1 ⊢ TrRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)}

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  {crab 3408   ⊆ wss 3916   class class class wbr 5109   ∘ ccom 5644   Rels crels 38166   TrRels ctrrels 38178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-rels 38471  df-ssr 38484  df-trs 38558  df-trrels 38559

This theorem is referenced by:  eltrrels3  38566