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Theorem eltrrels3 37071
Description: Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
eltrrels3 (𝑅 ∈ TrRels ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ 𝑅 ∈ Rels ))
Distinct variable group:   𝑥,𝑅,𝑦,𝑧

Proof of Theorem eltrrels3
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dftrrels3 37067 . 2 TrRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)}
2 breq 5112 . . . . . 6 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
3 breq 5112 . . . . . 6 (𝑟 = 𝑅 → (𝑦𝑟𝑧𝑦𝑅𝑧))
42, 3anbi12d 632 . . . . 5 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑦𝑟𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧)))
5 breq 5112 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑧𝑥𝑅𝑧))
64, 5imbi12d 345 . . . 4 (𝑟 = 𝑅 → (((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
762albidv 1927 . . 3 (𝑟 = 𝑅 → (∀𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
87albidv 1924 . 2 (𝑟 = 𝑅 → (∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
91, 8rabeqel 36743 1 (𝑅 ∈ TrRels ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107   class class class wbr 5110   Rels crels 36665   TrRels ctrrels 36677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-rels 36976  df-ssr 36989  df-trs 37063  df-trrels 37064
This theorem is referenced by: (None)
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