Theorem cnvelrels 38477

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	⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels )

Proof of Theorem cnvelrels

Step	Hyp	Ref	Expression
1		relcnv 6125	. 2 ⊢ Rel ◡𝐴
2		cnvexg 7947	. . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V)
3		elrelsrel 38469	. . 3 ⊢ (◡𝐴 ∈ V → (◡𝐴 ∈ Rels ↔ Rel ◡𝐴))
4	2, 3	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (◡𝐴 ∈ Rels ↔ Rel ◡𝐴))
5	1, 4	mpbiri 258	1 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels )

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2106  Vcvv 3478  ^◡ccnv 5688  Rel wrel 5694   Rels crels 38164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-rels 38467

This theorem is referenced by:  cosscnvelrels  38479