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Theorem dfcnvrefrel5 37272
Description: Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.)
Assertion
Ref Expression
dfcnvrefrel5 ( CnvRefRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑥 = 𝑦) ∧ Rel 𝑅))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem dfcnvrefrel5
StepHypRef Expression
1 dfcnvrefrel4 37271 . 2 ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))
2 cnvref5 37089 . 2 (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑥 = 𝑦)))
31, 2bianim 36965 1 ( CnvRefRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑥 = 𝑦) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wss 3945   class class class wbr 5142   I cid 5567  Rel wrel 5675   CnvRefRel wcnvrefrel 36921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-dm 5680  df-rn 5681  df-res 5682  df-cnvrefrel 37266
This theorem is referenced by:  dfantisymrel5  37501
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