Theorem elrefrelsrel 39249

Description: For sets, being an element of the class of reflexive relations (df-refrels 39240) 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 39091	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
2	1	anbi2d 641	. 2 ⊢ (𝑅 ∈ 𝑉 → ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅)))
3		elrefrels2 39247	. 2 ⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
4		dfrefrel2 39244	. 2 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2143   ∩ cin 3904   ⊆ wss 3905   I cid 5555   × cxp 5659  dom cdm 5661  ran crn 5662  Rel wrel 5666   Rels crels 38834   RefRels crefrels 38837   RefRel wrefrel 38838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-rels 39089  df-ssr 39227  df-refs 39239  df-refrels 39240  df-refrel 39241

This theorem is referenced by:  elrefsymrelsrel  39304