Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elcnvrefrelsrel Structured version   Visualization version   GIF version

Theorem elcnvrefrelsrel 35877
 Description: For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 35869) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
Assertion
Ref Expression
elcnvrefrelsrel (𝑅𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))

Proof of Theorem elcnvrefrelsrel
StepHypRef Expression
1 elrelsrel 35832 . . 3 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
21anbi2d 631 . 2 (𝑅𝑉 → ((𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ) ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)))
3 elcnvrefrels2 35875 . 2 (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))
4 dfcnvrefrel2 35873 . 2 ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
52, 3, 43bitr4g 317 1 (𝑅𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2115   ∩ cin 3918   ⊆ wss 3919   I cid 5446   × cxp 5540  dom cdm 5542  ran crn 5543  Rel wrel 5547   Rels crels 35560   CnvRefRels ccnvrefrels 35566   CnvRefRel wcnvrefrel 35567 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-rels 35830  df-ssr 35843  df-cnvrefs 35868  df-cnvrefrels 35869  df-cnvrefrel 35870 This theorem is referenced by:  elfunsALTVfunALTV  36035  eldisjsdisj  36065
 Copyright terms: Public domain W3C validator