Theorem brcosscnvcoss 38845

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 1863	. . 3 ⊢ (∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴))
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
3		brcoss 38842	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
4		brcoss 38842	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
5	4	ancoms 458	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
6	2, 3, 5	3bitr4d 311	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1781   ∈ wcel 2114   class class class wbr 5085   ≀ 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-coss 38822

This theorem is referenced by:  cocossss  38847  cnvcosseq  38848  rncossdmcoss  38866  symrelcoss3  38876  eleccossin  38894