Theorem cnvelrels 39699

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 6111	. 2 ⊢ Rel ◡𝐴
2		cnvexg 7957	. . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V)
3		elrelsrel 39565	. . 3 ⊢ (◡𝐴 ∈ V → (◡𝐴 ∈ Rels ↔ Rel ◡𝐴))
4	2, 3	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → (◡𝐴 ∈ Rels ↔ Rel ◡𝐴))
5	1, 4	mpbiri 261	1 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels )

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2147  Vcvv 3454  ^◡ccnv 5654  Rel wrel 5660   Rels crels 39308

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-rels 39563

This theorem is used by:  cosscnvelrels  39700