Theorem elcnvrefrelsrel 38533

Description: For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 38523) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)

Assertion

Ref	Expression
elcnvrefrelsrel	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))

Proof of Theorem elcnvrefrelsrel

Step	Hyp	Ref	Expression
1		elrelsrel 38484	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
2	1	anbi2d 630	. 2 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ) ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)))
3		elcnvrefrels2 38531	. 2 ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))
4		dfcnvrefrel2 38527	. 2 ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
5	2, 3, 4	3bitr4g 314	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ∩ cin 3902   ⊆ wss 3903   I cid 5513   × cxp 5617  dom cdm 5619  ran crn 5620  Rel wrel 5624   Rels crels 38177   CnvRefRels ccnvrefrels 38183   CnvRefRel wcnvrefrel 38184

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-rels 38482  df-ssr 38495  df-cnvrefs 38522  df-cnvrefrels 38523  df-cnvrefrel 38524

This theorem is referenced by:  elfunsALTVfunALTV  38695  eldisjsdisj  38725