Theorem dfcnvrefrel5 39737

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 39736	. 2 ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))
2		cnvref5 39475	. 2 ⊢ (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦)))
3	1, 2	bianim 587	1 ⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ⊆ wss 3900   class class class wbr 5106   I cid 5549  Rel wrel 5660   CnvRefRel wcnvrefrel 39316

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-cnvrefrel 39731

This theorem is used by:  dfantisymrel5  39989