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Theorem elrefrelsrel 34611
Description: The element of the class of all reflexive relations and the reflexive relation predicate are the same, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set. (Contributed by Peter Mazsa, 25-Jul-2021.)
Assertion
Ref Expression
elrefrelsrel (𝑅𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))

Proof of Theorem elrefrelsrel
StepHypRef Expression
1 elrelsrel 34579 . . 3 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
21anbi2d 614 . 2 (𝑅𝑉 → ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ) ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅)))
3 elrefrels2 34609 . 2 (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))
4 dfrefrel2 34607 . 2 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
52, 3, 43bitr4g 303 1 (𝑅𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wcel 2145  cin 3722  wss 3723   I cid 5157   × cxp 5248  dom cdm 5250  ran crn 5251  Rel wrel 5255   Rels crels 34317   RefRels crefrels 34320   RefRel wrefrel 34321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-rels 34577  df-ssr 34590  df-refs 34602  df-refrels 34603  df-refrel 34604
This theorem is referenced by:  elrefsymrelsrel  34659
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