Theorem eltrrels3 39346

Description: Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)

Assertion

Ref	Expression
eltrrels3	⊢ (𝑅 ∈ TrRels ↔ (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ 𝑅 ∈ Rels ))

Distinct variable group:   𝑥,𝑅,𝑦,𝑧

Proof of Theorem eltrrels3

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftrrels3 39342	. 2 ⊢ TrRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)}
2		breq 5113	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
3		breq 5113	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑧 ↔ 𝑦𝑅𝑧))
4	2, 3	anbi12d 644	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
5		breq 5113	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑧 ↔ 𝑥𝑅𝑧))
6	4, 5	imbi12d 347	. . . 4 ⊢ (𝑟 = 𝑅 → (((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
7	6	2albidv 1956	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
8	7	albidv 1953	. 2 ⊢ (𝑟 = 𝑅 → (∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
9	1, 8	rabeqel 38939	1 ⊢ (𝑅 ∈ TrRels ↔ (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ 𝑅 ∈ Rels ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146   class class class wbr 5111   Rels crels 38867   TrRels ctrrels 38879

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-rels 39122  df-ssr 39260  df-trs 39338  df-trrels 39339

This theorem is used by: (None)