Theorem elcnvrefrelsrel 38500

Description: For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 38490) 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 38451	. . 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 38498	. 2 ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))
4		dfcnvrefrel2 38494	. 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 2108   ∩ cin 3925   ⊆ wss 3926   I cid 5547   × cxp 5652  dom cdm 5654  ran crn 5655  Rel wrel 5659   Rels crels 38147   CnvRefRels ccnvrefrels 38153   CnvRefRel wcnvrefrel 38154

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-rels 38449  df-ssr 38462  df-cnvrefs 38489  df-cnvrefrels 38490  df-cnvrefrel 38491

This theorem is referenced by:  elfunsALTVfunALTV  38661  eldisjsdisj  38691