Theorem brin3 35140

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

Assertion

Ref	Expression
brin3	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆){{𝐵}}))

Proof of Theorem brin3

Step	Hyp	Ref	Expression
1		brin2 35139	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐵⟩))
2		opidg 4692	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ⟨𝐵, 𝐵⟩ = {{𝐵}})
3	2	adantl 474	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐵, 𝐵⟩ = {{𝐵}})
4	3	breq2d 4937	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐵⟩ ↔ 𝐴(𝑅 ⋉ 𝑆){{𝐵}}))
5	1, 4	bitrd 271	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆){{𝐵}}))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051   ∩ cin 3821  {csn 4435  ⟨cop 4441   class class class wbr 4925   ⋉ cxrn 34933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-fo 6191  df-fv 6193  df-1st 7499  df-2nd 7500  df-xrn 35105

This theorem is referenced by: (None)