Theorem cnvcosseq 35674

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 35671	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥))
2	1	el2v 3500	. . . . 5 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥)
3	2	biimpi 218	. . . 4 ⊢ (𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)
4	3	gen2 1790	. . 3 ⊢ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)
5		cnvsym 5967	. . 3 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥))
6	4, 5	mpbir 233	. 2 ⊢ ◡ ≀ 𝑅 ⊆ ≀ 𝑅
7		relcoss 35660	. . 3 ⊢ Rel ≀ 𝑅
8		relcnveq 35571	. . 3 ⊢ (Rel ≀ 𝑅 → (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅))
9	7, 8	ax-mp 5	. 2 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅)
10	6, 9	mpbi 232	1 ⊢ ◡ ≀ 𝑅 = ≀ 𝑅

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1528   = wceq 1530  Vcvv 3493   ⊆ wss 3934   class class class wbr 5057  ^◡ccnv 5547  Rel wrel 5553   ≀ ccoss 35445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-coss 35651

This theorem is referenced by:  br2coss  35675