Theorem eltrrelsrel 39863

Description: For sets, being an element of the class of transitive relations is equivalent to satisfying the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)

Assertion

Ref	Expression
eltrrelsrel	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ TrRels ↔ TrRel 𝑅))

Proof of Theorem eltrrelsrel

Step	Hyp	Ref	Expression
1		elrelsrel 39640	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
2	1	anbi2d 642	. 2 ⊢ (𝑅 ∈ 𝑉 → (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅)))
3		eltrrels2 39861	. 2 ⊢ (𝑅 ∈ TrRels ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
4		dftrrel2 39859	. 2 ⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ TrRels ↔ TrRel 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147   ⊆ wss 3900   ∘ ccom 5662  Rel wrel 5663   Rels crels 39383   TrRels ctrrels 39395   TrRel wtrrel 39396

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-rels 39638  df-ssr 39776  df-trs 39854  df-trrels 39855  df-trrel 39856

This theorem is used by: (None)