Theorem cnvelrels 36613

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 6012	. 2 ⊢ Rel ◡𝐴
2		cnvexg 7771	. . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V)
3		elrelsrel 36605	. . 3 ⊢ (◡𝐴 ∈ V → (◡𝐴 ∈ Rels ↔ Rel ◡𝐴))
4	2, 3	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (◡𝐴 ∈ Rels ↔ Rel ◡𝐴))
5	1, 4	mpbiri 257	1 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels )

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106  Vcvv 3432  ^◡ccnv 5588  Rel wrel 5594   Rels crels 36335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-rels 36603

This theorem is referenced by:  cosscnvelrels  36615