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

Theorem cnvelrels 38023
Description: The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019.)
Assertion
Ref Expression
cnvelrels (𝐴𝑉𝐴 ∈ Rels )

Proof of Theorem cnvelrels
StepHypRef Expression
1 relcnv 6103 . 2 Rel 𝐴
2 cnvexg 7930 . . 3 (𝐴𝑉𝐴 ∈ V)
3 elrelsrel 38015 . . 3 (𝐴 ∈ V → (𝐴 ∈ Rels ↔ Rel 𝐴))
42, 3syl 17 . 2 (𝐴𝑉 → (𝐴 ∈ Rels ↔ Rel 𝐴))
51, 4mpbiri 257 1 (𝐴𝑉𝐴 ∈ Rels )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  Vcvv 3463  ccnv 5671  Rel wrel 5677   Rels crels 37707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-rels 38013
This theorem is referenced by:  cosscnvelrels  38025
  Copyright terms: Public domain W3C validator