Theorem eltrrelsrel 39706

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 39483	. . 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 39704	. 2 ⊢ (𝑅 ∈ TrRels ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
4		dftrrel2 39702	. 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 2144   ⊆ wss 3896   ∘ ccom 5655  Rel wrel 5656   Rels crels 39226   TrRels ctrrels 39238   TrRel wtrrel 39239

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-rels 39481  df-ssr 39619  df-trs 39697  df-trrels 39698  df-trrel 39699

This theorem is used by: (None)