Theorem brin 5159

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 3930	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 5108	. 2 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ 𝑆))
3		df-br 5108	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5108	. . 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 2109   ∩ cin 3913  ⟨cop 4595   class class class wbr 5107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-in 3921  df-br 5108

This theorem is referenced by:  brinxp2  5716  trin2  6096  poirr2  6097  dfpo2  6269  predtrss  6295  tpostpos  8225  brinxper  8700  erinxp  8764  sbthcl  9063  infxpenlem  9966  fpwwe2lem11  10594  fpwwe2  10596  isinv  17722  isffth2  17880  ffthf1o  17883  ffthoppc  17888  ffthres2c  17904  isunit  20282  opsrtoslem2  21963  posrasymb  32891  trleile  32897  satefvfmla1  35412  brtxp  35868  idsset  35878  dfon3  35880  elfix  35891  dffix2  35893  brcap  35928  funpartlem  35930  trer  36304  fneval  36340  brcnvin  38352  brxrn  38356  brin2  38400  br1cossinres  38438  grumnud  44275