Theorem cnvelrels 38493

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 6078	. 2 ⊢ Rel ◡𝐴
2		cnvexg 7903	. . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V)
3		elrelsrel 38485	. . 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 2109  Vcvv 3450  ^◡ccnv 5640  Rel wrel 5646   Rels crels 38178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-rels 38483

This theorem is referenced by:  cosscnvelrels  38495