Theorem brin 5082

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 3897	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 5031	. 2 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ 𝑆))
3		df-br 5031	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5031	. . 3 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	3, 4	anbi12i 629	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
6	1, 2, 5	3bitr4i 306	1 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2111   ∩ cin 3880  ⟨cop 4531   class class class wbr 5030

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-br 5031

This theorem is referenced by:  brinxp2  5593  trin2  5950  poirr2  5951  tpostpos  7895  erinxp  8354  sbthcl  8623  infxpenlem  9424  fpwwe2lem12  10052  fpwwe2  10054  isinv  17022  isffth2  17178  ffthf1o  17181  ffthoppc  17186  ffthres2c  17202  isunit  19403  opsrtoslem2  20724  posrasymb  30670  trleile  30679  satefvfmla1  32785  dfpo2  33104  brtxp  33454  idsset  33464  dfon3  33466  elfix  33477  dffix2  33479  brcap  33514  funpartlem  33516  trer  33777  fneval  33813  brxrn  35786  brin2  35817  br1cossinres  35847  grumnud  40994