Theorem brcosscnvcoss 38891

Description: For sets, the 𝐴 and 𝐵 cosets by 𝑅 binary relation and the 𝐵 and 𝐴 cosets by 𝑅 binary relation are the same. (Contributed by Peter Mazsa, 27-Dec-2018.)

Assertion

Ref	Expression
brcosscnvcoss	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴))

Proof of Theorem brcosscnvcoss

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exancom 1868	. . 3 ⊢ (∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴))
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
3		brcoss 38888	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
4		brcoss 38888	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
5	4	ancoms 459	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
6	2, 3, 5	3bitr4d 312	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∃wex 1786   ∈ wcel 2119   class class class wbr 5072   ≀ ccoss 38550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-coss 38868

This theorem is referenced by:  cocossss  38893  cnvcosseq  38894  rncossdmcoss  38912  symrelcoss3  38922  eleccossin  38940