Theorem brin 4694

Description: The intersection of two relations. (Contributed by FL, 7-Oct-2008.)

Assertion

Ref	Expression
brin	⊢ (A(R ∩ S)B ↔ (ARB ∧ ASB))

Proof of Theorem brin

Step	Hyp	Ref	Expression
1		elin 3220	. 2 ⊢ (⟨A, B⟩ ∈ (R ∩ S) ↔ (⟨A, B⟩ ∈ R ∧ ⟨A, B⟩ ∈ S))
2		df-br 4641	. 2 ⊢ (A(R ∩ S)B ↔ ⟨A, B⟩ ∈ (R ∩ S))
3		df-br 4641	. . 3 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
4		df-br 4641	. . 3 ⊢ (ASB ↔ ⟨A, B⟩ ∈ S)
5	3, 4	anbi12i 678	. 2 ⊢ ((ARB ∧ ASB) ↔ (⟨A, B⟩ ∈ R ∧ ⟨A, B⟩ ∈ S))
6	1, 2, 5	3bitr4i 268	1 ⊢ (A(R ∩ S)B ↔ (ARB ∧ ASB))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ∩ cin 3209  ⟨cop 4562   class class class wbr 4640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-br 4641

This theorem is referenced by:  brinxp2  4836  brres  4950  intasym  5029  fncnv  5159  dfid4  5504  trtxp  5782  brtxp  5784  elfix  5788  ersymtr  5933  porta  5934  sopc  5935  weds  5939  enpw1lem1  6062  enmap2lem1  6064  enmap1lem1  6070  nchoicelem8  6297  nchoicelem19  6308