Theorem cnvelrels 38451

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 6134	. 2 ⊢ Rel ◡𝐴
2		cnvexg 7964	. . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V)
3		elrelsrel 38443	. . 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 2108  Vcvv 3488  ^◡ccnv 5699  Rel wrel 5705   Rels crels 38137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-rels 38441

This theorem is referenced by:  cosscnvelrels  38453