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Theorem dftrrel3 36792
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 36791 . 2 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
2 cotr 6030 . . 3 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
32anbi1i 625 . 2 (((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅))
41, 3bitri 275 1 ( TrRel 𝑅 ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1537  wss 3892   class class class wbr 5081  ccom 5604  Rel wrel 5605   TrRel wtrrel 36396
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-trrel 36788
This theorem is referenced by:  trrelressn  36797  dfeqvrel3  36805
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