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Theorem cnvelrels 35914
 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 5935 . 2 Rel 𝐴
2 cnvexg 7614 . . 3 (𝐴𝑉𝐴 ∈ V)
3 elrelsrel 35906 . . 3 (𝐴 ∈ V → (𝐴 ∈ Rels ↔ Rel 𝐴))
42, 3syl 17 . 2 (𝐴𝑉 → (𝐴 ∈ Rels ↔ Rel 𝐴))
51, 4mpbiri 261 1 (𝐴𝑉𝐴 ∈ Rels )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2111  Vcvv 3441  ◡ccnv 5519  Rel wrel 5525   Rels crels 35634 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-dm 5530  df-rn 5531  df-rels 35904 This theorem is referenced by:  cosscnvelrels  35916
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