Theorem cnvcosseq 39154

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 39151	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥))
2	1	el2v 3462	. . . . 5 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥)
3	2	biimpi 219	. . . 4 ⊢ (𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)
4	3	gen2 1826	. . 3 ⊢ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)
5		cnvsym 6116	. . 3 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥))
6	4, 5	mpbir 234	. 2 ⊢ ◡ ≀ 𝑅 ⊆ ≀ 𝑅
7		relcoss 39140	. . 3 ⊢ Rel ≀ 𝑅
8		relcnveq 38955	. . 3 ⊢ (Rel ≀ 𝑅 → (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅))
9	7, 8	ax-mp 5	. 2 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅)
10	6, 9	mpbi 233	1 ⊢ ◡ ≀ 𝑅 = ≀ 𝑅

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  Vcvv 3455   ⊆ wss 3906   class class class wbr 5110  ^◡ccnv 5662  Rel wrel 5668   ≀ ccoss 38810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-coss 39128

This theorem is referenced by:  br2coss  39155