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Theorem elcnvrefrelsrel 35652
Description: For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 35644) 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 35607 . . 3 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
21anbi2d 628 . 2 (𝑅𝑉 → ((𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ) ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)))
3 elcnvrefrels2 35650 . 2 (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))
4 dfcnvrefrel2 35648 . 2 ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
52, 3, 43bitr4g 315 1 (𝑅𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  cin 3932  wss 3933   I cid 5452   × cxp 5546  dom cdm 5548  ran crn 5549  Rel wrel 5553   Rels crels 35336   CnvRefRels ccnvrefrels 35342   CnvRefRel wcnvrefrel 35343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-rels 35605  df-ssr 35618  df-cnvrefs 35643  df-cnvrefrels 35644  df-cnvrefrel 35645
This theorem is referenced by:  elfunsALTVfunALTV  35810  eldisjsdisj  35840
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