Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cosselrels Structured version   Visualization version   GIF version

Theorem cosselrels 37879
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 37802 . 2 (𝐴𝑉 → ≀ 𝐴 ∈ V)
2 relcoss 37806 . . 3 Rel ≀ 𝐴
3 elrelsrel 37870 . . 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 3468  Rel wrel 5674  ccoss 37556   Rels crels 37558
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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
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 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-coss 37794  df-rels 37868
This theorem is referenced by:  cosscnvelrels  37880  dffunsALTV2  38067  dffunsALTV3  38068  dffunsALTV4  38069  elfunsALTV2  38076  elfunsALTV3  38077  elfunsALTV4  38078  elfunsALTV5  38079
  Copyright terms: Public domain W3C validator