Theorem dftrrels3 38574

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 38573	. 2 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
2		cotr 6086	. 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 3917   class class class wbr 5110   ∘ ccom 5645   Rels crels 38178   TrRels ctrrels 38190

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 5254  ax-nul 5264  ax-pr 5390

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 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-rels 38483  df-ssr 38496  df-trs 38570  df-trrels 38571

This theorem is referenced by:  eltrrels3  38578