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Theorem cosselrels 37822
Description: Cosets of sets are elements of the relations class. Implies (𝑅 ∈ Rels → ≀ 𝑅 ∈ Rels ). (Contributed by Peter Mazsa, 25-Aug-2021.)
Assertion
Ref Expression
cosselrels (𝐴𝑉 → ≀ 𝐴 ∈ Rels )

Proof of Theorem cosselrels
StepHypRef Expression
1 cossex 37745 . 2 (𝐴𝑉 → ≀ 𝐴 ∈ V)
2 relcoss 37749 . . 3 Rel ≀ 𝐴
3 elrelsrel 37813 . . 3 ( ≀ 𝐴 ∈ V → ( ≀ 𝐴 ∈ Rels ↔ Rel ≀ 𝐴))
42, 3mpbiri 258 . 2 ( ≀ 𝐴 ∈ V → ≀ 𝐴 ∈ Rels )
51, 4syl 17 1 (𝐴𝑉 → ≀ 𝐴 ∈ Rels )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3466  Rel wrel 5671  ccoss 37499   Rels crels 37501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-coss 37737  df-rels 37811
This theorem is referenced by:  cosscnvelrels  37823  dffunsALTV2  38010  dffunsALTV3  38011  dffunsALTV4  38012  elfunsALTV2  38019  elfunsALTV3  38020  elfunsALTV4  38021  elfunsALTV5  38022
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