Theorem cnvcosseq 37098

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 37095	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥))
2	1	el2v 3480	. . . . 5 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥)
3	2	biimpi 215	. . . 4 ⊢ (𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)
4	3	gen2 1798	. . 3 ⊢ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)
5		cnvsym 6101	. . 3 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥))
6	4, 5	mpbir 230	. 2 ⊢ ◡ ≀ 𝑅 ⊆ ≀ 𝑅
7		relcoss 37084	. . 3 ⊢ Rel ≀ 𝑅
8		relcnveq 36982	. . 3 ⊢ (Rel ≀ 𝑅 → (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅))
9	7, 8	ax-mp 5	. 2 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅)
10	6, 9	mpbi 229	1 ⊢ ◡ ≀ 𝑅 = ≀ 𝑅

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   = wceq 1541  Vcvv 3472   ⊆ wss 3943   class class class wbr 5140  ^◡ccnv 5667  Rel wrel 5673   ≀ ccoss 36834

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5141  df-opab 5203  df-xp 5674  df-rel 5675  df-cnv 5676  df-coss 37072

This theorem is referenced by:  br2coss  37099