Theorem cosscnvelrels 38484

Description: Cosets of converse sets are elements of the relations class. (Contributed by Peter Mazsa, 31-Aug-2021.)

Assertion

Ref	Expression
cosscnvelrels	⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ Rels )

Proof of Theorem cosscnvelrels

Step	Hyp	Ref	Expression
1		cnvelrels 38482	. 2 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels )
2		cosselrels 38483	. 2 ⊢ (◡𝐴 ∈ Rels → ≀ ◡𝐴 ∈ Rels )
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ Rels )

Syntax hints:   → wi 4   ∈ wcel 2109  ^◡ccnv 5618   ≀ ccoss 38165   Rels crels 38167

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 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-coss 38398  df-rels 38472

This theorem is referenced by:  dfdisjs2  38697  eldisjs2  38711