Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inres2 Structured version   Visualization version   GIF version

Theorem inres2 35552
 Description: Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.)
Assertion
Ref Expression
inres2 ((𝑅𝐴) ∩ 𝑆) = ((𝑅𝑆) ↾ 𝐴)

Proof of Theorem inres2
StepHypRef Expression
1 inres 5844 . . 3 (𝑆 ∩ (𝑅𝐴)) = ((𝑆𝑅) ↾ 𝐴)
21ineqcomi 35551 . 2 ((𝑅𝐴) ∩ 𝑆) = ((𝑆𝑅) ↾ 𝐴)
3 incom 4153 . . 3 (𝑅𝑆) = (𝑆𝑅)
43reseq1i 5822 . 2 ((𝑅𝑆) ↾ 𝐴) = ((𝑆𝑅) ↾ 𝐴)
52, 4eqtr4i 2847 1 ((𝑅𝐴) ∩ 𝑆) = ((𝑅𝑆) ↾ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∩ cin 3909   ↾ cres 5530 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-rab 3135  df-v 3473  df-in 3917  df-res 5540 This theorem is referenced by:  xrnres  35696
 Copyright terms: Public domain W3C validator