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Theorem dftrrel3 38180
Description: Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
dftrrel3 ( TrRel 𝑅 ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅))
Distinct variable group:   𝑥,𝑅,𝑦,𝑧

Proof of Theorem dftrrel3
StepHypRef Expression
1 dftrrel2 38179 . 2 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
2 cotr 6117 . . 3 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
32anbi1i 622 . 2 (((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅))
41, 3bitri 274 1 ( TrRel 𝑅 ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wss 3944   class class class wbr 5149  ccom 5682  Rel wrel 5683   TrRel wtrrel 37794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-trrel 38176
This theorem is referenced by:  trrelressn  38185  dfeqvrel3  38193
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