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Theorem symrelcoss2 35586
Description: The class of cosets by 𝑅 is symmetric, see dfsymrel2 35665. (Contributed by Peter Mazsa, 27-Dec-2018.)
Assertion
Ref Expression
symrelcoss2 (𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Proof of Theorem symrelcoss2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symrelcoss3 35585 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)
2 cnvsym 5967 . . 3 (𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
32anbi1i 623 . 2 ((𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅))
41, 3mpbir 232 1 (𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526  wss 3933   class class class wbr 5057  ccnv 5547  Rel wrel 5553  ccoss 35334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-coss 35539
This theorem is referenced by:  symrelcoss  35676
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