Theorem brin 5124

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 3899	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 5073	. 2 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ 𝑆))
3		df-br 5073	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5073	. . 3 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	3, 4	anbi12i 634	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
6	1, 2, 5	3bitr4i 304	1 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∈ wcel 2119   ∩ cin 3882  ⟨cop 4561   class class class wbr 5072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-in 3890  df-br 5073

This theorem is referenced by:  brinxp2  5696  trin2  6073  poirr2  6074  dfpo2  6247  predtrss  6273  tpostpos  8186  brinxper  8663  erinxp  8728  sbthcl  9027  infxpenlem  9926  fpwwe2lem11  10555  fpwwe2  10557  isinv  17718  isffth2  17876  ffthf1o  17879  ffthoppc  17884  ffthres2c  17900  isunit  20344  opsrtoslem2  22032  zsoring  28419  posrasymb  33046  trleile  33050  satefvfmla1  35653  brtxp  36106  idsset  36116  dfon3  36118  elfix  36129  dffix2  36131  brcap  36166  funpartlem  36170  trer  36544  fneval  36580  brcnvin  38745  brxrn  38750  brin2  38805  br1cossinres  38904  grumnud  44730