Theorem dfcnvrefrel5 38534

Description: Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.)

Assertion

Ref	Expression
dfcnvrefrel5	⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem dfcnvrefrel5

Step	Hyp	Ref	Expression
1		dfcnvrefrel4 38533	. 2 ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))
2		cnvref5 38352	. 2 ⊢ (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦)))
3	1, 2	bianim 576	1 ⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ⊆ wss 3951   class class class wbr 5143   I cid 5577  Rel wrel 5690   CnvRefRel wcnvrefrel 38191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-cnvrefrel 38528

This theorem is referenced by:  dfantisymrel5  38763