Theorem brin 5141

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 3913	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 5090	. 2 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ 𝑆))
3		df-br 5090	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5090	. . 3 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	3, 4	anbi12i 628	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
6	1, 2, 5	3bitr4i 303	1 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2111   ∩ cin 3896  ⟨cop 4579   class class class wbr 5089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3904  df-br 5090

This theorem is referenced by:  brinxp2  5692  trin2  6069  poirr2  6070  dfpo2  6243  predtrss  6269  tpostpos  8176  brinxper  8651  erinxp  8715  sbthcl  9012  infxpenlem  9904  fpwwe2lem11  10532  fpwwe2  10534  isinv  17667  isffth2  17825  ffthf1o  17828  ffthoppc  17833  ffthres2c  17849  isunit  20291  opsrtoslem2  21991  zsoring  28332  posrasymb  32948  trleile  32952  satefvfmla1  35469  brtxp  35922  idsset  35932  dfon3  35934  elfix  35945  dffix2  35947  brcap  35982  funpartlem  35984  trer  36358  fneval  36394  brcnvin  38406  brxrn  38410  brin2  38454  br1cossinres  38492  grumnud  44327