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Theorem elrefrelsrel 37028
Description: For sets, being an element of the class of reflexive relations (df-refrels 37019) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
Assertion
Ref Expression
elrefrelsrel (𝑅𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))

Proof of Theorem elrefrelsrel
StepHypRef Expression
1 elrelsrel 36995 . . 3 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
21anbi2d 630 . 2 (𝑅𝑉 → ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ) ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅)))
3 elrefrels2 37026 . 2 (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))
4 dfrefrel2 37023 . 2 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
52, 3, 43bitr4g 314 1 (𝑅𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  cin 3910  wss 3911   I cid 5531   × cxp 5632  dom cdm 5634  ran crn 5635  Rel wrel 5639   Rels crels 36682   RefRels crefrels 36685   RefRel wrefrel 36686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-rels 36993  df-ssr 37006  df-refs 37018  df-refrels 37019  df-refrel 37020
This theorem is referenced by:  elrefsymrelsrel  37079
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