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Theorem elcnvrefrelsrel 38517
Description: For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 38507) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
Assertion
Ref Expression
elcnvrefrelsrel (𝑅𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))

Proof of Theorem elcnvrefrelsrel
StepHypRef Expression
1 elrelsrel 38468 . . 3 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
21anbi2d 630 . 2 (𝑅𝑉 → ((𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ) ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)))
3 elcnvrefrels2 38515 . 2 (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))
4 dfcnvrefrel2 38511 . 2 ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
52, 3, 43bitr4g 314 1 (𝑅𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2105  cin 3961  wss 3962   I cid 5581   × cxp 5686  dom cdm 5688  ran crn 5689  Rel wrel 5693   Rels crels 38163   CnvRefRels ccnvrefrels 38169   CnvRefRel wcnvrefrel 38170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-rels 38466  df-ssr 38479  df-cnvrefs 38506  df-cnvrefrels 38507  df-cnvrefrel 38508
This theorem is referenced by:  elfunsALTVfunALTV  38678  eldisjsdisj  38708
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