Theorem symrelcoss2 38430

Description: The class of cosets by 𝑅 is symmetric, see dfsymrel2 38513. (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 38429	. 2 ⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅)
2		cnvsym 6073	. . 3 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥))
3	2	anbi1i 624	. 2 ⊢ ((◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) ↔ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅))
4	1, 3	mpbir 231	1 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ⊆ wss 3911   class class class wbr 5102  ^◡ccnv 5630  Rel wrel 5636   ≀ ccoss 38142

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 5246  ax-nul 5256  ax-pr 5382

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-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-coss 38375

This theorem is referenced by:  symrelcoss  38524