Theorem symrelcoss2 38923

Description: The class of cosets by 𝑅 is symmetric, see dfsymrel2 39000. (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.

Step	Hyp	Ref	Expression
1		symrelcoss3 38922	. 2 ⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅)
2		cnvsym 6064	. . 3 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥))
3	2	anbi1i 630	. 2 ⊢ ((◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) ↔ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅))
4	1, 3	mpbir 232	1 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   ⊆ wss 3883   class class class wbr 5072  ^◡ccnv 5617  Rel wrel 5623   ≀ ccoss 38550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-coss 38868

This theorem is referenced by:  symrelcoss  39011