Theorem brin2 34661

Description: Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.)

Assertion

Ref	Expression
brin2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐵⟩))

Proof of Theorem brin2

Step	Hyp	Ref	Expression
1		brxrn 34630	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐵⟩ ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵)))
2	1	3anidm23 1545	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐵⟩ ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵)))
3		brin 4895	. 2 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵))
4	2, 3	syl6rbbr 282	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐵⟩))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∈ wcel 2157   ∩ cin 3768  ⟨cop 4374   class class class wbr 4843   ⋉ cxrn 34468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-1st 7401  df-2nd 7402  df-xrn 34627

This theorem is referenced by:  brin3  34662