Theorem brcoss3 39122

Description: Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.)

Assertion

Ref	Expression
brcoss3	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ([𝐴]◡𝑅 ∩ [𝐵]◡𝑅) ≠ ∅))

Proof of Theorem brcoss3

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brcnvg 5869	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ V) → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴))
2	1	elvd 3468	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴))
3		brcnvg 5869	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑢 ∈ V) → (𝐵◡𝑅𝑢 ↔ 𝑢𝑅𝐵))
4	3	elvd 3468	. . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝐵◡𝑅𝑢 ↔ 𝑢𝑅𝐵))
5	2, 4	bi2anan9 649	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴◡𝑅𝑢 ∧ 𝐵◡𝑅𝑢) ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
6	5	exbidv 1949	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝐴◡𝑅𝑢 ∧ 𝐵◡𝑅𝑢) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
7		ecinn0 38952	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]◡𝑅 ∩ [𝐵]◡𝑅) ≠ ∅ ↔ ∃𝑢(𝐴◡𝑅𝑢 ∧ 𝐵◡𝑅𝑢)))
8		brcoss 39120	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
9	6, 7, 8	3bitr4rd 315	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ([𝐴]◡𝑅 ∩ [𝐵]◡𝑅) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1807   ∈ wcel 2150   ≠ wne 2965  Vcvv 3462   ∩ cin 3912  ∅c0 4294   class class class wbr 5114  ^◡ccnv 5664  [cec 8695   ≀ ccoss 38782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ec 8699  df-coss 39100

This theorem is referenced by:  br2coss  39127  trcoss2  39173