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Theorem dftrrels3 36313
Description: Alternate definition of the class of transitive relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
Assertion
Ref Expression
dftrrels3 TrRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)}
Distinct variable group:   𝑥,𝑟,𝑦,𝑧

Proof of Theorem dftrrels3
StepHypRef Expression
1 dftrrels2 36312 . 2 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}
2 cotr 5946 . 2 ((𝑟𝑟) ⊆ 𝑟 ↔ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))
31, 2rabbieq 36013 1 TrRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1540   = wceq 1542  {crab 3057  wss 3843   class class class wbr 5030  ccom 5529   Rels crels 35958   TrRels ctrrels 35970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-rels 36226  df-ssr 36239  df-trs 36309  df-trrels 36310
This theorem is referenced by:  eltrrels3  36317
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