Theorem cnvcosseq 38848

Description: The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.)

Assertion

Ref	Expression
cnvcosseq	⊢ ◡ ≀ 𝑅 = ≀ 𝑅

Proof of Theorem cnvcosseq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brcosscnvcoss 38845	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥))
2	1	el2v 3436	. . . . 5 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥)
3	2	biimpi 216	. . . 4 ⊢ (𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)
4	3	gen2 1798	. . 3 ⊢ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)
5		cnvsym 6077	. . 3 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥))
6	4, 5	mpbir 231	. 2 ⊢ ◡ ≀ 𝑅 ⊆ ≀ 𝑅
7		relcoss 38834	. . 3 ⊢ Rel ≀ 𝑅
8		relcnveq 38649	. . 3 ⊢ (Rel ≀ 𝑅 → (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅))
9	7, 8	ax-mp 5	. 2 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅)
10	6, 9	mpbi 230	1 ⊢ ◡ ≀ 𝑅 = ≀ 𝑅

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542  Vcvv 3429   ⊆ wss 3889   class class class wbr 5085  ^◡ccnv 5630  Rel wrel 5636   ≀ ccoss 38504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-coss 38822

This theorem is referenced by:  br2coss  38849