Theorem dfcnvrefrel4 38471

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

Assertion

Ref	Expression
dfcnvrefrel4	⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))

Proof of Theorem dfcnvrefrel4

Step	Hyp	Ref	Expression
1		df-cnvrefrel 38466	. 2 ⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2		cnvref4 38289	. 2 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ I ))
3	1, 2	bianim 576	1 ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∩ cin 3923   ⊆ wss 3924   I cid 5544   × cxp 5649  dom cdm 5651  ran crn 5652  Rel wrel 5656   CnvRefRel wcnvrefrel 38129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-br 5117  df-opab 5179  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-cnvrefrel 38466

This theorem is referenced by:  dfcnvrefrel5  38472  dfantisymrel4  38700