Theorem elrefrelsrel 39512

Description: For sets, being an element of the class of reflexive relations (df-refrels 39503) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)

Assertion

Ref	Expression
elrefrelsrel	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))

Proof of Theorem elrefrelsrel

Step	Hyp	Ref	Expression
1		elrelsrel 39354	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
2	1	anbi2d 642	. 2 ⊢ (𝑅 ∈ 𝑉 → ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅)))
3		elrefrels2 39510	. 2 ⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
4		dfrefrel2 39507	. 2 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145   ∩ cin 3898   ⊆ wss 3899   I cid 5545   × cxp 5649  dom cdm 5651  ran crn 5652  Rel wrel 5656   Rels crels 39097   RefRels crefrels 39100   RefRel wrefrel 39101

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 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-rels 39352  df-ssr 39490  df-refs 39502  df-refrels 39503  df-refrel 39504

This theorem is used by:  elrefsymrelsrel  39567